The question is important, as it’s often used as an argument against idea of immortality, on the level of desirability as well as feasibility. It may result in less interest in radical life extension as "result will be the same", we will die. Religion, on the other hand is not afraid to "sell" immortality, as it has God, who will solve all contradiction in immortality implementation. As a result, religion win on the market of ideas.
Immortality (by definition) is about not dying. The fact of eternal linear existence follows from it, seems to be very simple and obvious theorem:
“If I do not die in the time moment N and N+1, I will exist for any N”.
If we prove that immortality is impossible, then any life would look like: Now + unknown very long time + death. So, death is inevitable, and the only difference is the unknown time until it happens.
It is an unpleasant perspective, by the way.
So we have or “bad infinity”, or inevitable death. Both look unappealing. Both also look logically contradictory. "Infinite linear existence" requires infinite memory of observer, for example. "Death of observer" is also implies an idea of the ending of stream of experiences, which can't be proved empirically, and from logical point of view is unproved hypothesis.
But we can change our point of view if we abandon the idea of linear time.
Physics suggests that near black holes closed time-like curves could be possible. https://en.wikipedia.org/wiki/Closed_timelike_curve (Idea of "Eternal recurrence" of Nietzsche is an example of such circle immortality.)
If I am in such a curve, my experiences may recur after, say, one billion years. In this case, I am immortal but have finite time duration.
It may be not very good, but it is just a starting point in considerations that would help lead us away from the linear time model.
There may be other configurations in non-linear time. Another obvious one is the merging of different personal timelines.
Another is the circular attractor.
Another is a combination of attractors, merges and circular timelines, which may result in complex geometry.
Another is 2 (or many)- dimensional time, with another perpendicular time arrow. It results in a time topology. Time could also include singularities, in which one has an infinite number of experiences in finite time.
We could also add here idea of splitting time in quantum multiverse.
We could also add an idea that there is a possible path between any two observer-moment, and given that infinitely many such paths exist in splitting multiverse, any observer has non zero probability to become any other observer, which results in tangle of time-like curves in the space of all possible minds.
Timeless physics ideas also give us another view on idea of “time” in which we don’t have “infinite time”, but not because infinity is impossible, but because there is no such thing as time.
TL;DR: The idea of time is so complex that we can’t state that immortality results in eternal linear existence. These two ideas may be true or false independently.
Also I have a question to the readers: If you think that superintelligence will be created, do you think it will be immortal, and why?