As requested, a puzzle post! Straight from the WTF-variant department. Quite hard.
This is a Fillomino, with the additional constraint that for each polyomino, there must not exist a path (i.e. a sequence of cells, each orthogonally adjacent to the next) that includes each of the polyomino’s cells exactly once (and does not include cells outside the polyomino).
As a degenerate case, 1-ominoes are banned as well.