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.

EDIT: the example below does not answer the question because the map is not smooth. (I had not read the question carefully, sorry!). I don't remove the answer since it might still be useful to somebody.

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.