Is the localization of a complete $A$-module $B$ complete? (Where $(A,\mathfrak{m})$ is $\mathfrak{m}$-adically complete)

Question

Let $(A, \mathfrak{m})$ be a local noetherian ring and let $A \subset B$ be a finite $A$-algebra. Assume that $A$ is $\mathfrak{m}$-adically complete. It follows that $B$ is $\mathfrak{m}$-adically complete viewed as an $A$-module. Let $\mathfrak{m}_1,\dots,\mathfrak{m}_n$ be the maximal ideals of $B$ lying over $\mathfrak{m}B$. Is it true that $B_{\mathfrak{m}_i}$ is $\mathfrak{m}$-adically complete? I would need it to show that $B \cong \prod_{i=1}^nB_{\mathfrak{m_i}}$.