Is every local quasi-finite morphism finite?

Question

Let $A\to B$ be a finite morphism, $(A,\mathfrak m)$ be a local ring, and $\mathfrak n$ be a maximal ideal of $B$ lying over $\mathfrak m$. Is there an example such that $A\to B_{\mathfrak n}$ is not finite?

Answer 1

$R\to S^{-1}R$ integral $\implies$ $S\subseteq U(R)$.

If $A\to B_{\mathfrak n}$ is finite then $B\to B_{\mathfrak n}$ is integral, and therefore $B-\mathfrak n\subseteq U(B)$, so $\bigcup_{\mathfrak n'\in\text{Max}(B)} \mathfrak n'\subseteq \mathfrak n$. If $B$ is not local this is impossible. Now I leave you the pleasure to find a concrete example.