If a sheaf is zero on every fiber is it non-zero

Question

Let $\pi:\mathcal{X} \to S$ be a flat morphism between projective varieties and $E$ be a coherent sheaf on $\mathcal{X}$ (not necessarily flat over $S$). If $E_s:=E|_{\mathcal{X}_s}$ is zero (meaning has no support) for any $s \in S$ then is it possible that $E$ is non-zero i.e., has non-trvial support?