If a sheaf is coherent on proper fibers, is the pushforward coherent?

Question

Suppose $f : X \to Y$ is a proper morphism of varieties. Let $F$ be a quasicoherent sheaf on $X$ which is coherent on the fibers ($F|_{f^{-1}(y)}$ is coherent for all geometric points $y$, or similar condition). Is it the case that $f_* F$ is coherent on $Y$?

This easily reduces to the case when $Y$ is affine. Under some hypothesis one could try to apply the "cohomology and base-change" type theorems, but those all assume coherence of $F$ anyway...

Answer 1

Take $F=k(X)$, the field of rational functions on $X$ with $f:X\to Y=X$, the identity map.