It's been maybe twenty years since I read this, so please forgive me for not remembering if it was a novel or a short story.
It's the story of a girl whose father was either a Greek or Egyptian alchemist. He had concocted an elixir which, if drunk periodically, grants immortality. The problem was that in order for it to work, the person drinking it couldn't have gone through puberty.
He gave the elixir to his son and daughter, and taught them how to make more. The children live through many centuries, but the son eventually stops taking it. He ages normally and eventually dies. The daughter continues to take it, and the reader learns her story while she is being interviewed, either by MDs, or psychiatrists.