Yes, this is possible.

Here are some real-life examples:

Woman carries out caesarean on herself - and she and baby live

A pregnant woman in Mexico is reported to have given birth to a
healthy baby boy after performing a caesarean section on herself with
a kitchen knife...The unidentified 40-year-old [other sources give her name as Ines Ramirez Perez], who lives in a rural
area without electricity, running water or sanitation, and whose home
was an eight-hour drive from the nearest hospital, performed the
operation when she found she could not deliver the baby naturally.

She took three small glasses of hard liquor and, using a kitchen
knife, sliced her abdomen in three attempts ... and delivered a male
infant who breathed immediately and cried...Before losing
consciousness the woman told one of her children to call a local nurse
for help. After the nurse stitched the wound with a sewing needle and
cotton thread, the mother and baby were transferred and treated...at
the nearest hospital."

Doctor rescued from Antarctica in 1999 dies at 57

Dr. Jerri Nielsen...treated herself for breast cancer while stationed
at the South Pole in 1999...Nielsen caught the nation's attention in
1999, when she found a lump in her breast as a 47-year-old physician
stationed at the Amundsen-Scott South Pole Research Station. After
finding the lump in June, she diagnosed herself with breast cancer and
began treating herself using chemotherapy agents that the U.S. Air
Force parachuted to the station the next month.

It was later revealed...that Nielsen -- an emergency room doctor from
Cleveland, Ohio -- performed a biopsy on herself with the help of
non-medical crew, who practiced using needles on a raw chicken.

In your character's case, the surgery is something from the future so hopefully, the tools are too. She'd be unlikely to be able to reach her spine properly but even today we have robots that do surgery and can be controlled by computer. So she could do that.

Imagine robotic surgery, for example (and this is modern day; in the OP's world, the tools would be higher tech).

And imagine an operating version of this:

Is it possible to perform surgery on yourself?

Leonid Rogozov, when based on a Russian Antarctic base as doctor, was forced to perform appendix operation on himself.

On the morning of 29 April 1961, Rogozov experienced general weakness, nausea, and moderate fever, and later pain in the lower right portion of the abdomen. None of the possible conservative treatment measures helped. By 30 April signs of localised peritonitis became apparent, and his condition worsened considerably by the evening. Mirny, the nearest Soviet research station, was more than 1,600 km (1,000 mi) from Novolazarevskaya. Antarctic research stations of other countries did not have an aircraft. Severe blizzard conditions prevented aircraft landing in any case. Rogozov had no option but to perform the operation on himself.

The operation started at 02:00 local time on 1 May with the help of a driver and meteorologist, who were providing instruments and holding a mirror to observe areas not directly visible, while Rogozov was in a semi-reclining position, half-turned to his left side. A solution of 0.5% novocaine was used for local anaesthesia of the abdominal wall. Rogozov made a 10–12 cm incision of the abdominal wall, and while opening the peritoneum he accidentally injured the cecum and had to suture it. Then he proceeded to expose the appendix. According to his report, the appendix was found to have a dark stain at its base, and Rogozov estimated it would have burst within a day. The appendix was resected and antibiotics were applied directly into the peritoneal cavity. General weakness and nausea developed about 30–40 minutes after the start of the operation so that short pauses for rest were repeatedly needed after that. By about 04:00 the operation was complete.

After the operation gradual improvement occurred in the signs of peritonitis and in the general condition of Rogozov. Body temperature returned to normal after five days, and the stitches were removed seven days after the operation. He resumed his regular duties in about two weeks.

So, it seems that is should be possible, with adequate preparation, to perform self surgery.

Is it possible to perform surgery by yourself on your spine?

But in the above case the abdomen is accessible with the hands with no big hurdles. Accessing the back region is going to be more cumbersome, especially considering the unnatural posture of the hands and the elbow when operating on oneself's back, if the operation has to be executed by hands.

If instead your character has the necessary tools for performing tele-surgery (basically remotely controlled robotic arms and hands), it should be feasible.

The visor needs to be adapted to a laying position and of course the subject cannot undergo total anesthesia, but only local anesthesia shall be applied. Moreover, considering that the back will be cut open to access the spine, movement of the arms shall be prevented and the tools shall be operated only by finger movement.

All in all, it's pretty unrealistic. Then again, if it's SciFi, why not.

Even the documented real cases where people have (allegedly) done surgery on themselves are for the most part somewhere in between "fucking heroic" and "unbelieveable".

I am not going to say it's all lies, but as someone who performed that kind of intervention (on others, in proper conditions) for a living 15 years ago, the idea of e.g. operating your own appendix seems like something that's, well let's just say not very likely to succeed unless you are some kind of superhuman (or credible, for that matter).

Taking Rogozov as an example. With nothing but local anaesthetic, your belly will be hard as rock as soon as you cut in, let alone touch the peritoneum (with muscle relaxant on the other hand side, good luck operating or breathing). Speaking of which... the sensation of touching, or cutting through your peritoneum (which is not under local anaesthetic) is a sensation worth having. Imagine being on a football field, and one of those meatbags taking a 20-meter charge, then kicking you in the crown jewels, as hard as he can. That's approximately the feeling of cutting in your peritoneum. Staying conscious and actually doing something that requires delicate manual work for an entire 20 minutes after that? Well, kudos on being a real tough guy, Mr. Rogozov. But I'm more inclined to believe this is a case of Korsakoff, no offense intended.

Also it's worth to note that local anaesthetics are ineffective in acidic environment such as... oh wait, an inflammated appendix. Then there's that thing about accidentially cutting open the caecum and suturing it. And yes, not only living to tell, but going to work again after only two weeks. I've seen 22 year olds spending a week struggling for their lives with peritonitis in the ICU after having one of those incidents. And that was in the early 2000s, consuming a 5-digit sum of care and antibiotics. This the kind of incident that, no matter how big your balls are, gives you cold sweat when you're an experienced surgeon. Let alone someone in 1960 who graduated a year earlier and is doing the operation on himself lying on his back. I mean, sure, when you're desperate and have few choices you probably try everything, but seriously... that's so fucking awesome superhuman that its factual accurracy is hard to believe.

But even if we assume the accounts are totally true and accurate -- the fact that one person did it and survived does not mean it is "doable" (in a sense that you can do it and expect to live). That's like saying Mrs. Vulović survived a fall from 10,000 meters (when in reality, she didn't fall at all), so it's doable to fall without parachute from 10,000 meters. Well, go ahead and try, no worries, you will always arrive on the ground.

As for the question's details:

clean stable environment and there is no risk of infection
surgery near her spine [...] inserting a microchip connecting to the nerve connected to the brain

So basically, cut through the skin and open the liquid-filled spinal canal which rules out "no risk of infection".
Then insert a foreign body that connects to a single nerve connected to the brain. Well, far down in the cauda that would probably even be doable (if this still counts as "connected to the brain", which is technically not the case, being the lower motoneuron) but further up the spine where you find the real "connected to brain" neurons, good luck cutting open the bundle and fiddling with it without accidentially pulling an axon or two (or ten) and provoking a jerk with devastating results.

All that over a monitor via remote manipulators (unless you are a Chinese acrobat of sorts?). Uh. Well sure, while at it, why not transplant your head onto a different body.

Unless you consider some very, very tough SciFi voodoo (AI assisted half-robotic remote manipulators with, I don't know... force fields, super awesome nanobots, engineered bacteria that keep the skin barrier but refuse to grow in spinal fluid, bacta-tanks that magically cure everything, whatever?) I don't think this can be done, realistically. Not with any chance of success anyway.

It sounds like the most difficult part would be actually reaching around to to insert the microchip if it's behind her own back. With a local anaesthetic, she'd be able to make the incision and insert the chip. Obviously I can't speak to the extent of the futuristic technology, or how involved the surgery is, but a simple insertion certainly wouldn't be too hard.

You might be interested in reading about a Russian doctor called Leonid Rogozov. He was the lone physician on an Antarctic expedition when he developed appendicitis. With no other options he gave himself a local anaesthetic and performed a self-appendectomy.

In his case he did have assistance in the form of a driver and meteorologist, but he made a full recovery and resumed his duties about a week later.

Another related topic would be biohackers or grinders, which is a term for people who experiment with cybernetic implants upon themselves. An example would be inserting small silicon-coated magnets under the fingertips, which can give the person a primitive electromagnetic sense as the magnet twitches and pushes against the nerve endings in the presence of an electrical field.

So if the technology is futuristic then why should she perform surgery herself. Let the future create a smart surgical pod that is capable of doing any kind of surgery. Just select a surgery and proceed and the computer is smart enough to finish the surgery with perfection.

I remember the surgery scene from the film Prometheus.

Using current technology I agree with the answers saying that this is not possible.

My suggestion would be to have the chips specially made and equipped with nano-technology for this purpose though. All she has to do is make a cut near her spine and slide the chip in. It will automatically activate once inside and extend nano-fibres to penetrate the spine and connect itself.

This makes the required surgery much simpler and safer to the point that it becomes a lot more believable.