They could take damage I guess? They're invincible, not invulnerable, so you could lose limbs until you can't move anymore, but don't die, like the black knight from monty python!
The player doesn't have to be the thing that's invincible. If you go with that idea though, there could still be lose conditions. Getting trapped somewhere, even when invincible, keeps you trapped there. Falling off a mountain you're climbing makes you lose your progress if you're invincible. The theme is pretty broad in that sense 👍
