Let $f:X \rightarrow Y$ be a flat surjective morphism of projective varieties with connected fibers, $X$ normal, $Y$ smooth. Suppose $L \in Pic (X)$ is nef such that $L|_{X_{y}} \cong \mathcal{O}_{X_{y}}$ for all $y \in Y$. Then does there exist $L_Y \in Pic(Y)$ such that $ L \cong f^*(L_Y)$?

We can see that $f_*(L) \in Pic(Y)$ by Grauert's theorem. In case all the fibers are integral, it is easy to show that $L \cong f^*f_*(L)$ (none of this needs nefness). However in general, integrality of all fibers need not be satisfied and I'm not sure how to argue.