I have done functional programming (FP) with Haskell (including professionally) for years, and I think this notion is misguided.

One of the core principles of FP is about state. However, I believe it is to think very carefully about state and be very explicit about how you handle it rather than performing ad-hoc mutations. That often means avoiding state, but many problems require maintaining state.

State machines can be a great tool to be explicit about state management to the point that you can have a specification for how state behaves.