Dying in Dungeons and Dragons 5th Edition very rarely occurs by chance. Importantly being reduced to 0 HP only renders a player character unconscious.
Finishing off Unconscious PCs
Finishing off Unconscious PCs
An intelligent enemy will likely notice if their targets keep getting healed by a party member. At that point they may ensure they "Finish the job". Each successful attack within 5 ft against an unconscious player causes them 2 death save failures. Meaning one of the assassins in your scenario (assuming they are the Monster Manual specific NPC named Assassin and that you are not using the name as a general description) could use their turn, attack the unconscious player twice, and kill them.
Typically most DMs will not immediately attack unconscious player characters. However if the Monsters in your encounter have witnessed the player character coming back up from being knocked out, this could be an indicator they need to go for the kill.
In addition to intelligent enemies recognizing players are returning, many enemies may make sense to have some sort of feeding frenzy on a knocked out player character. Zombies, rabid animals, swarms, crocodiles, quipper, sharks and some monstrosities may be unable to resist the urge of feeding on fresh food even in the middle of combat.
Also remember that if a damage source reduces your character to negative their hit point total, the character is instantly killed. This makes bringing players back into the fight with a few hit points particularly dangerous if you have hard hitting abilities.
Hit Point Maximum Reduction
Hit Point Maximum Reduction
If healing in particular is giving you trouble, you can consider some of the monsters which have maximum hit points reduction. These include : Blue Slaad, Diseased Rats, Specter, Bulezau, Chasme, Wraith, Wight, Clay Golem, Succubus and others.
Ability Score Damage
Ability Score Damage
A few creatures have ability score damage, such as Shadow or Intellect devourer. These give a sense of danger which cannot be immediately remedied in combat.
Long Term Status Effects
Long-Term Status Effects
Long term status effects are also harder to fix. Famously the Medusa, Gorgon, Cocatrice, and Basilisk can turn foes to stone. A slightly less permanent version is the many incapacitating poisons. Giant Spiders, Giant Centipede, Giant Wasp, or Phase Spiders are good examples.
Splitting the Party
Splitting the Party
Another way of ensuring a more difficult encounter is to have monsters attempt to ambush players when they are separated. They can also use their abilities to force players apart so they are unable to support one another. Many creatures have grapple abilities, which after used, they can move away from the party with their prey. They may do this with conscious or unconscious players. This is especially effective if a monster has a method of travel that is unavailable to the players. Climbing, flying, or swimming could allow a creature to get away faster than the players can give chase.
Conclusion
Conclusion
Hopefully between these methods you will find a way to make your party feel a sense of danger in their encounters. You should probably have these used against an NPC rather than a player character to demonstrate their danger first.