Let $A$ be a commutative noetherian ring, $I$-adically complete (and separated) with respect to an ideal $I \subseteq A$.
Let $M$ be a finite $A$-module, and let $N$ be an $I$-adically complete $A$-module.
Is it true that $M\otimes_A N$ is also $I$-adically complete?
If $M$ is free this is clear. Is it true in general?