Let $Y\subset \mathbb A^3$ be the quadric cone, defined by $xy-z^2=0$ and  take the map $X={\mathbb A}^2\to Y$  given by $(u,v)\mapsto (u^2,v^2, uv)$. This is a (non flat) double cover and the direct image of ${\mathcal O}_X$ is of the form ${\mathcal O}_Y\oplus F$, where $F$ is a rank  1 reflexive sheaf that is not locally free.