Injective $\mathcal O_X$-module which is not an injective sheaf

Question

Can you find an example of a ringed space $(X,\mathcal O_X)$ and an injecitve $\mathcal O_X$-module (i.e. an injective object in the category $\mathfrak{Mod}(X)$ of $\mathcal O_X$-modules) which is not an injective sheaf (i.e. an injective object in the category $\mathfrak{Ab}(X)$ of sheaves of abelian groups on $X$)?

This question is related to this one.

Answer 1

A nonzero injective vector space over $\mathbb{F}_2$ is not an injective abelian group.

So just take $X$ to be a single point, with $\mathcal{O}_X(X)=\mathbb{F}_2$.