Suppose that $A$ is a commutative noetherian Jacobson ring and $M$ is an $A$-module. Suppose in addition that $M$ is $\mathfrak{m}$-adically separated for every maximal ideal $\mathfrak{m}$, and that $M/\mathfrak{m}M$ is $A/\mathfrak{m}$-finite for every $\mathfrak{m}$. Does it follow that $M$ is $A$-finite?
If I imposed no separatedness conditions, then $\mathbf{Q}$ as a $\mathbf{Z}$-module would be a counterexample. If I did not require $A$ to be Jacobson, or if I only imposed a separatedness condition at one maximal ideal, then $k[\![t]\!]$ as a $k[t]$- or $k[t]_{(t)}$-module would be a counterexample.
If it helps, I'm happy to assume that the $\mathfrak{m}$-adic completions $M_{\mathfrak{m}}^\wedge$ of $M$ are all finite projective of the same rank. The rings I have in mind are polynomial rings over fields, or (strict) affinoid algebras over discretely valued fields, so I'm also happy to make additional assumptions (such as excellence) on $A$.