Reproduction is not physically special. If it is physically possible to create a new organism with a longer lifespan than its parent (which it is), then it is physically possible to extend the lifespan of an existing organism. This may be impractical, but we cannot prove that no organism can live ≈forever, because it's not a physical impossibility. (Assuming a source of power remains present: it looks like the universe won't last forever, so we can trivially assert that no organism can live forever.)
> it is physically possible to extend the lifespan of an existing organism.
This is not at all guaranteed until we actually manage to do it. And that's exactly what we're discussing, you can't just say "but it must be possible". There's no rule of the universe that says it should, and given life has been around for 4 billion years and there's no single species (specially animals which is what we really should consider here if we're talking about human lifespan) that manages to live for more than a few hundred years, I think that's strong evidence that life is approaching a fundamental limit here. Someone else said "maybe that's enough" - well, why?? Maybe 100 years is already quite enough then??
Take the existing organism apart, fix all the problems, put the organism back together. No laws of physics prevent this, so by the totalitarian principle it's physically possible. Principles of engineering might make it impractical, and even if it is practical we might never discover how to do it, but it's possible.