Modules over Completion

I have the following question.

Let $R$ be a commutative ring with unit, and let $\hat{R}$ denote its completion (w.r.t. any ideal $I$). Let $M$ be an $\hat{R}$-module. Is $M= N\otimes_R \hat{R}$ for some $R$-module $N$?

Let $A \rightarrow B$ be a faithfully flat morphism of rings, then the category of $A$-modules is equivalent to the category of $B$-modules with descent data (see here). In less precise words, a $B$-module is of the form $M \otimes_A B$ exactly when one can provide this descent data.

When $\hat{A}$ is the completion of a noetherian ring $A$ at an ideal $I$ contained in the Jacobson radical, then the map $A \rightarrow \hat{A}$ is faithfully flat. For example, this is satisfied for any noetherian local ring.

I'm not sure if it is reasonable to expect a more complete answer (i.e. not Noetherian or $I \not\subset \text{Jac}(A)$).