I don't think we could, but for philosophic reasons, rather than technical ones.
For something to be declared "immortal," is a high claim. But how do you decide if something is immortal?
There is a famous thought experiment called the Ship of Theseus, from ancient Greek times. The Ship of Theseus, in the story, is a famous warship responsible for countless victories. However, over the years, she's needed repairs. Replacement ribs, replacement rudders, etc. Eventually there isn't a single piece remaining that has seen one of her famous victories. Is she still the Ship of Theseus? If not, when did she cease to be?
One would need to pick a stance about what it means for a "person" to be alive before one could dare talk about immortality. Honestly, the Ship of Theseus is still used today, because there is no consensus as to the answer. Strange phrases like "endurable" and "perdurable," that you rarely see in every day life, crop up trying to explain this dilemma. Literally speaking, what might be immortal to one, thanks to massive replacement of hardware may be dead to another.
The other issue is entropy. Lots of people translate "immortal" as "living a lot longer than they can think about." However, this can become very difficult as you get onto longer scales. Stephen Baxter's book, Manifold Time, explores these longer time scales. Sure, everything could be pretty peachy for a while, but what about when the sun goes out? On a cosmic scale, that's actually going to happen pretty soon (only about 4 billion years away). If your plan is to escape to other stars, what happens when they all go out? Surprisingly, there's a lot more life left in the universe after the stars go out. What about proton decay? Immortality is tough when the particles we think of as defining existence start to wink out.
These issues make defining "immortal" remarkably difficult. Thus, you won't find just one answer to your problem.
Consider how many religions declare us to already be immortal in one way or another. Buddhism has their system of rebirth. Christianity has their eternal soul. Daoism says we've always been part of the immortal, yet ephemeral. Maybe we already have our immortality, we're just looking in the wrong place.