Unwinnable by Design: In the Unholy Cathedral, you are completely incapable of firing with any weapon that's not a melee weapon, and fight the Angel of Death at the end, an extremely powerful boss that has as much HP as the Spider Mastermind, insanely high armor (the Mastermind and demonic spiders of the game have an armor rating of 2, while the Angel has a rating of 10), and a very powerful melee attack with high accuracy that can kill you in as little as three hits. Assuming you aren't running a melee build, if you have not invested in melee boosting traits (particularly the Berserker trait) or have Hatred Skulls (to gain berserk), and do not have a decent melee weapon, you have no hope of killing this boss if you were not lucky enough to come into the level while invulnerable with a nuke.
An earlier bonus level, The Wall, is also unwinnable if you did not reach the level with a Rocket Launcher and enough ammunition, as you won't be able to get past the wall if you are not able to get lucky with a Phase device to teleport past it.
A later bonus level, The Vaults, require you to use the Arena Master's staff to open the vaults. If you do not have it though, you can still open two of the vaults, by finding and destroying the one destroyable wall tile in the vault's wall. The other vault however, cannot be open by any other means, thus unless you are to get lucky with a Phase device to teleport inside it, you have no way of defeating the enemies and collecting the items inside, preventing you from truly completing the level.
Nothing in Unwinnable by Design is correct. That's for rendering the game unwinnable, not being able to complete the level. In all cases, you can still leave the level through the stairs, even if you don't accomplish anything. I'm leaving them here, as they probably do apply for some other trope, though I don't know which one.
The Unwinnable by Mistake entry I'm similarly suspicious of, as that doesn't sound quite right for the trope. I could just be picky, though, as several similar examples are on the other Unwinnable pages.
Don't take life too seriously. It's only a temporary situation.
Removed the following for misuse.
Nothing in Unwinnable by Design is correct. That's for rendering the game unwinnable, not being able to complete the level. In all cases, you can still leave the level through the stairs, even if you don't accomplish anything. I'm leaving them here, as they probably do apply for some other trope, though I don't know which one.
The Unwinnable by Mistake entry I'm similarly suspicious of, as that doesn't sound quite right for the trope. I could just be picky, though, as several similar examples are on the other Unwinnable pages.
Don't take life too seriously. It's only a temporary situation.