Solution by technicality: they're still in combat
A (combat) encounter is only resolved when the danger has abated and the task has been resolved. If someone is bleeding out, they're still in danger. Ergo, the encounter has not completed, and it continues on. In particular, the combat is ongoing because the threat posed by the opponents has not ceased. This is similar to any effects (magical or mundane) which persist after the evident opponents are killed or driven off. If there's a blade barrier still going, or an alchemist's fire burning, that would affect a player where he is standing, he still needs to get out of that area and survive it. Same issue with the dying character. The combat only ends once all opponents are defeated and all threats to the party members have been neutralized. So keep running the combat until this has been achieved.
People really can die in a matter of seconds
But you may think "oh, the only threat is bleeding out, and it seems silly to let a guy die in a matter of seconds when there's nothing else threatening him." Well, this is exactly what happens in the real world. We don't have magical healing spells, but we have pretty amazing technology and training (in first world countries, at least), and even with doctors and/or EMTs actively trying to help the person they can still die in a matter of seconds. It is not unrealistic in the slightest to maintain that this character is in dire need of assistance and is straddling the line between life and death.
Death is legitimately hard to stop (sans magic)
Keep this in mind when considering whether to let a player "take 10" on a check to help the dying character stabilize. This is normally not something you're allowed to do when there is a penalty for failure—a failed disarm check triggers the trap, for example—or for tasks which you cannot casually take your time at. Trying to save a dying person is not something you can deal with in such a casual fashion even with advanced medical technology. Time is of the essence, and while a professional knows how to keep a cool head and a steady hand, that doesn't eliminate the need to act quickly.
It's D&D. Death is cheap.
So then you might think "oh, but a character dying is such a horrible thing! That's much too drastic!"
Unless you are using a homebrew set of rules or other very specific/custom setting, death is not a big deal for a party of adventurers. Resurrection spells are available relatively early, and there are likely several friendly NPC clerics that are high enough level that would be willing to do the resurrection (at cost, unless previous adventures they've done have left the cleric in their debt). In some settings it is reasonable to expect that there are accessible NPCs with access to the most powerful resurrection spells, which come with essentially no drawbacks.
A dead PC is not an end. It's an adventure hook!
Death is not meant to be the end of that adventurer's tale in D&D. It's meant to be a part of it, which he can continue on from. And what DM doesn't find themselves wanting for organic (rather than pre-planned and forced) ways to advance a story and send the players out on another adventure? Bringing back a dead companion is usually one of the best motivations for a party. If they can't do the resurrection themselves, or really want to use one of the higher level resurrections they don't otherwise have access to: use it to your advantage! Now's a great time to introduce a religion/character/cleric/organization/outsider/whatever that can fix their problem, and will be important later on.