#mathlib has a fascinating set of counterexamples, which includes, for example, a disproof of the Aharoni–Korman conjecture: https://leanprover-community.github.io/mathlib4_docs/Counterexamples/AharoniKorman.html
when i'm in a #lean4 phase, i often think of (much, much more trivial) false statements, so i decided to just start proving that they were indeed false: https://g.pyrope.net/simple-counterexamples
the pace of additions will be somewhat rapid because the proofs are only a couple of lines at most. they're fun to think about, though—how would you prove that (haskell syntax) Monad m => m a -> a is inconsistent?