Here is a question that I'm just copying from Math Stack Exchange that I asked awhile ago. It has just been sitting there unanswered, and although I haven't really thought about it since I posted it, I'm still very interested in a nice example if it exists.

Suppose $(\mathfrak{X}, \mathcal{O}_\mathfrak{X})$ is a Noetherian formal scheme and let $\mathcal{I}$ be an ideal of definition. Then we have a system of schemes $X_n=(|\mathfrak{X}|, \mathcal{O}_\mathfrak{X}/\mathcal{I}^n)$.

If the inverse system $\Gamma (X_n, \mathcal{O}_{X_n})\to \Gamma (X_{n-1},\mathcal{O}_{X_{n-1}})$ satisfies the Mittag-Leffler condition (the images eventually stabilize), then we get some particularly nice properties such as $Pic(\mathfrak{X})=\lim Pic(X_n)$.

More generally, we don't have to be worried about converting between thinking about coherent sheaves on the formal scheme and thinking about them as compatible systems of coherent sheaves on actual schemes.

My question:

Is there a known example of a formal scheme for which that system of global sections does not satisfy the Mittag-Leffler condition?

One thing to note is that it can't be affine (the maps are all surjective) or projective (finite dimensionality forces the images to stabilize).

A subquestion is whether or not there is a general reason to believe such an example exists. People I talk to usually say things along the lines of: you definitely have to be careful here because in principle this could happen. But no one seems to have ever thought up an example.

Lastly (still related...I think), is there a known example where you can't think of coherent (or maybe invertible) sheaves as systems because the two aren't the same?