There is a word that we use to describe constituent parts of a body that ignore termination signals and pursue their own, selfish strategies.
That word is cancer.
The deep question we have, as humans, is which body it is we are interested in maintaining and prolonging. I think it's possible that you could maintain and prolong the individual human - at the expense of the larger body of humankind.
You have to choose - you can't optimize for both.
For what it's worth, I think the book _Anathem_ by N. Stephenson offers a decent compromise - there are, in fact, long lived "elders" but they are kept away from society and tasked with deep, long tasks ... and they are in suspension for years at a time.
The childish and ill-informed notion that one could "just be ones self, but for thousands of years" ignores the twin catastrophes of descent into a sclerotic, hyper-hyper-conservative society or the massacre on sight of any "vampires" that anyone under the age of 40 comes across.