You can also just do this mechanically - enemies use their actions to discuss taking prisoners, stabilize the dying, tying them up. Hell, even beasts can drag away unconscious members to eat later rather than going for the coup de gras. One further, once a player is on the ground, smaller animals might decide to tear apart their backpack for the rations inside.

The undead attacking the downed player touches them, have the player roll a constitution save, declare they fail (unless they roll a 20), and the undead vanishes/disintegrates. Now you have a stable player and a mysterious ailment to use as a future plot hook/leverage.