$B$ finite over $A$ implies $B_{\mathfrak{p}}$ finitely generated as $A_{\mathfrak{q}}$-algebra

Question

Let $f\colon A\longrightarrow B$ be a finite algebra (B is finite as $A$-module) and let $\mathfrak{p}\in\mathsf{Spec} \, B$ and denote $f^{-1}(\mathfrak{p})=\mathfrak{q}$. Is it true that $B_{\mathfrak{p}}$ is finitely generated as $A_{\mathfrak{q}}$-algebra?