Personality definitely has a hefty genetic component, and the purely psychological side is likely quite self-reinforcing - nervous people find things to be nervous around, gregarious people encounter other gregarious people telling them it is great to be together, the optimist remembers all the times things turned out good, and so on. It is however not entirely fixed,
people shift a bit in their "big 5" personality scales as they age (in particular agreeableness and conscientousness increase while neuroticism decrease as people mature).
I would expect really old people to have a relatively settled personality style, but a lot of the surface expressions and current values may shift. Big events and cultural changes can make a lot of difference too - across a very long life you are bound to encounter a number of life-changing events just by chance.
A good fictional example of a long life causing personality shifts is the protagonist of Bruce Sterling's
Schismatrix. He goes from a being a revolutionary to theatre manager, space pirate, diplomat, officer in the academic-military complex, extortionist, invalid, prophet and explorer across his multi-century life.
I like the idea of the doctor's skill mattering for keeping you on track. Bad treatments may make your identity too fluid or lock you in, a good doctor can get nerve growth to happen in the right places and ensure sensible memory and personality continuity. ("Oh, Dr Antar is the best! He managed to keep my husband's memories of our early dating perfectly clear; we celebrated our adamantium anniversary last decade!")
It also raises the possibility of "plasticity murder": deliberately making someone's brain regenerate a lot, blurring their old identity and personality and essentially turning them into teenagers again. ("Yes, Lord Andronicus is still in charge and owns the company, but he is not quite the staid proconsul you knew; in fact, I think you will find his political opinions a bit... distasteful.")
Long marriages, that is another interesting problem. Married couples are said to become more similar to each other over time. Imagine the effects of a multi-century marriage! Of course, it might be that natural love does not last that long. But there might be ways of fixing that neurochemically, see (shameless plug)
my paper on the subject. ("Dr Antar has been keeping my wife nicely affectionate for 50 years now. Of course, she doesn't know.")