I wondered about that too! It's possible the writers aren't sure either and don't want to have to deal with the question...
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Given the amount of concern the Scions always show when someone's in a life-threatening position, I would assume they can. I mean, obviously they don't know *for certain* as no one's actually died, but it's a fair bet that they can. It should be noted that while their bodies aren't on the First, their souls very much so are. If dying on the First means their soul dies / is destroyed, then that is as good as death.
In support of this, I'm fairly certain it was mentioned pre-ShB that the healthy states of the Scions bodies indicated that their souls were still yet living somewhere else, as if their souls had perished, the body would die as well.
Rewatching the cutscenes, Thancred was engaging Ran'jit with the intention of it being a last stand, and was dying after - he was just fortunate that Urianger had detoured to find the twins and was catching up with the others later, so they passed through the same area and found him before he actually died.