The FHI just released a technical report, Indefinite Survival through Backup Copies whose result is:

it is possible to have a finite chance to survive an infinite time even if there is a finite chance of getting destroyed per unit of time, if you make backup copies (that are also destroyable) at a high enough rate. The number of backup copies needed only grows logarithmically with time, a surprisingly slow growth.

I'd previously been assuming that exponentially many copies were needed (in order to "cancel out" the fact that if you only have one copy you die exponentially quickly), so finding out that only logarithmically many copies are needed is great news. I no longer need to hope that space is negatively curved!