Actually, so far as I've seen, based on dominion specific modifiers on prophets and such, it seems to treat each combatant as if they had never left the province they are in before the fight. I want to say that I saw an immortal unit survive a loss in the arena, but it was some months back when I actually paid any attention to the death matches.
But I do recall very clearly seeing pretenders and prophets with boosted HP, as if they were still in their own dominion. Small chance that I happened to only look at the ones whose dominion actually housed the arena for that match, but that seems highly unlikely to me.