Short Version:
Maybe P is overwhelmed by bookkeeping and it's distracting him from situational awareness. Help him make a mechanically very simple character without fiddly bits or conditionals to keep track of, so he can focus on making good choices rather than having good bookkeeping. Invite the other players to support P with advice and by being good role models for the behaviour he's trying to cultivate.
Long form answer, with rambling and details.
Back in my very first RPG ever--and also my first time as a GM--I had a player whose poor choices got him repeatedly killed. Let's call him Q.
Q knew the rules and mechanics quite well, but had a very hard time applying them intelligently to whatever situation he found himself in (like forgetting to heal himself as a cleric). Even more than that, though, was his role-playing: he really really liked to role-play his characters, but that got him in trouble because when Q got deep into his character's internal motives the PC would lose common sense and perspective about the surrounding context of his actions.
It got bad. Really bad. Q's second character was killed by the party for betraying them (he had a conversation about his friends over tea with a "nice" lady). At that point I shared Making the Tough Decisions with the group. He studied it carefully, had intense discussions with me about it... and as a direct result his fourth character perished of untempered curiosity: the characterisation "very curious" overcame the common sense "half these items are cursed and my friends are begging me to stop," until the pile of treasure he was investigating yielded up a lethal curse.
After that session I took Q aside and we talked. He knew he had a problem, and he was trying to "get better," but he needed help. I'd noticed that all his PCs so far were mechanically complicated and required in-game bookkeeping: advanced casters and races with lots of conditional features and spell-like abilities to keep track of. So we hatched the simplest possible character build: nothing to keep track of. No "if you're flanking, X also happens," no spells, no per-day abilities. If his character sheet said he could do a thing, he could always do it.
We wound up with a kind of Indiana Jones flavoured skillmonkey (a rogue chassis with homebrew mods to replace things like sneak attack because tracking whether you can deal that extra damage was beyond what we wanted for the build). He wasn't optimised in the traditional sense--but since another PC in the party had straight levels in the NPC Expert class, that wasn't an issue in keeping him relevant in the group. Instead he was optimised for what Q needed: a simple no-bookkeeping character to let him focus on situational awareness and making good choices.
At the end of each session he'd hang back --along with any other players who wanted to-- and we'd reflect on the game: what worked, what didn't. We'd consult (and if necessary research) and come up with what to make sure we did again, and what we'd change next time. (I've since found that any game I run which has some form of this "reflect and plan" dynamic after every session is improved by it.)
In tandem with another player rising to the challenge and being a kind of "teach by example" role model, it worked. A year later Q was successfully running complicated wizard builds with great party dynamics and great depth of character. He was a real joy to work with, and all he needed was to wade in at the shallow end of the bookkeeping pool instead of jumping into the deepest part head-first.
nota bene: My players have tended to treat the group dynamic as one of table-level cooperation between friends. However much their characters may be rivals, at the table they collaborate to tell the best stories, and I'm also one of the collaborators. In groups where players and/or the GM act as rivals at the table level of things, I'm not sure how much my experience will be useful. It sounds like your whole group isn't really on the same page in terms of their desired gameplay experience, and communication isn't really strong. Working on improving the "friends at the table" level of things might help your game in a number of ways.
The easiest way to gain advantage on a death saving throw is to use inspiration.
If
you
have
inspiration,
you
can
expend
it
when
you
make
an
attack
roll,
saving
throw,
or
ability
check.
Spending
your
inspiration
gives
you
advantage
on
that
roll.
Alternatively, you could be a Wild Magic Sorcerer, and use Tides of Chaos. Well, probably.
Starting at 1st level, you can manipulate the forces of
chance and chaos to gain advantage on one attack roll,
ability check, or saving throw.
The 3rd- level spell Beacon of Hope explicitly grants advantage on death saving throws for its duration.
Fighters can use Indomitable to reroll a failed saving throw - not precisely the same as advantage, but pretty close.
Beginning at 9th level, you can reroll a saving throw that
you fail.
There are almost certainly more options that I've missed - getting advantage on things is a pretty fundamental part of 5e.
Best Answer
As a preliminary note, a good solution to this problem will (as usual) involve talking to your players. Ask them if they think this is even a problem, and if so, what kind. You may find that they enjoy being reckless and don't mind dealing with the consequences, even if that means inaction or even death for their characters. If so, congratulations: you're all set! But if they don't want to be unconscious and bored, read on.
You can try different things to keep them entertained, like having them run NPCs, narrate the combat to make it more epic, or take notes for posterity. If your players are flexible, that may work, and they may even enjoy the change of pace. (Though honestly, in my experience once they're stabilized it's usually a good time for a bio break to get more drinks for everyone or take care of other necessities, which should resolve the situation nicely unless it's a really long fight.)
However, most players are at the table because they want to play their characters. (Not all of them, but most of the standard motivations require being able to influence the game in some capacity, and most of those involve investment in a specific character.) If that's true, then no substitute will ultimately be as satisfying as being in there with your own character, mixing it up.
So if I could back up a bit, I would ask a different question, namely, why is this situation occurring so frequently that you have to ask?
If PC's routinely get KO'd, I suspect one of three things is true: A) your PC's are low level; B) your PC's are employing bad strategies; or C) your encounters are structured in a way that causes a lot of KO's, possibly because you're using Glass Cannons (warning: TVTropes).
A) should resolve itself pretty quickly. B) might resolve itself as your players learn better strategies, but you may need to ask if they want any pointers. If it's C), though, you can fix this by adjusting your tactics, both in building encounters and how you run them. A plan failing because "You didn't even dent her armor!" can be as spooky as one failing because they spotted you and stabbed you. (Of course, if you go too far with defensive enemies, fights can drag on forever, but sometimes people have to be defeated in a particular way that requires a little in-game research or experimentation.)
More importantly: if your boss fights are deadly and/or you've successfully cultivated an atmosphere where they feel deadly, when a PC does drop your players should be scrambling to get them back up so they don't lose the advantage of numbers (not to mention, y'know, keeping them alive). You've said the party doesn't have a ton of healing, but you can influence that by pointing them towards Goodberry and other useful spells and abilities, adding an NPC, or just giving out healing potions as loot (something that's working pretty well for 2 groups I'm in with relatively little healing). If people are watching each other's backs, a PC shouldn't be down for more than a round or so.
If you take the above into consideration, I think you can probably settle into a rhythm where this isn't too much of a problem in the first place.