Do you make death saving throws out of combat?

Death saving throws are made on each turn where you are dying and not stable. However, the rules specifically talk about making saves during combat rounds – so what about after combat? Two PCs are up, three are dying, and the lich is defeated. For the three that are dying:

  1. Do combat turns continue as normal until all are either stable are dead?
  2. Since combat is over, do the dying PCs make their saves consecutively until they are dead/stable without interference from other PCs? (a combat round in which a save is normally made is only 6 seconds)
  3. Do the PCs automatically stabilize?

I think the closest interpretation is option 1, but at my home game we’ve house-ruled to work with option 3 (barring exigent circumstances like a PC dying alone somewhere in a trap) because we assume that in the majority of cases a party can easily stabilize their companion or the PC will make their save.

I’m looking for a rule stating explicitly how death saves work outside of combat, or a wiser interpretation than my own.