Let $k$ be an algebraically closed field of characteristic 0. Let $X$ be a smooth and connected variety over $k$, with $k[X]^\ast = k^\ast$. Suppose that $f: Y \to X$ is a surjective finite etale map of $k$-varieties, with $Y$ connected as well.
Is it true that $k[Y]^\ast = k^\ast$?
As a remark: I don't want to make properness assumptions on $X$ or $Y$ (properness would certainly imply the above).