Oh, it isn't that I don't want things to die, more just I don't want every combat to resolve into killing most things.
And that's fair enough.
Thing is, without greatly rejigging how D&D runs its hit point system you're somewhat swimming against the tide on this one.
What you'd need is:
--- a much "wider" area between fully functional and dead, for PCs and living monsters* alike, such that it becomes way more possible to knock something out without killing it
--- rules to handle unconsciousness: how it happens, when it happens, the odds of it happening, that sort of thing
--- rules to handle revival from unconsciousness: what types of cures work, how effective they are, etc.
--- resting rules for after someone's been knocked out
The low-hanging design space for adding in that "wider area" is negative hit points. You could, say have it that a creature taken to or below 0 h.p. risks going unconscious (and automatically goes unconscious if taken below a lower threshold - say, -20) but doesn't die unless a) the hit that sent it below 0 was a critical or b) it reaches [negative its full h.p. total] or c) it was coup-de-graced regardless of its hit point status.
Characters at or below 0 h.p. but still conscious are at disadvantage to everything and advantage cannot be gained.
Cures received if at or below 0 h.p. are at half effect. If after a cure the recipient is above -20 it automatically wakes up; if still at or below -20 it remains unconscious. An untended unconscious creature at or below -20 will die after [some random amount of time - d6 hours?]; an untended unconscious creature above -20 can roll [need some mechanics here] to wake up, stay asleep, or eventually die.
* - non-living monsters e.g. undead and constructs would stay at all-or-nothing - they're either functional or they're not, no middle ground. Were it me I'd also put oozes and jellies into this category as well.
Yeah, it is a big one and I don't want to take it off the table.
@vincegetorix has some good idea, as do others, and for the most part the more I think about it I like the idea of "dying" only on a critical hit, otherwise 0 hp will be defeat (surrender, flee, etc.).