Lately I have become interested in solid $F$-modules where $F$ is some discrete field. Ideally, one would want a category that is as nicely behaved as solid abelian groups or solid $\mathbb{F_p}$-modules.
I have managed to show that a discrete field is a solid module over itself, so we get that the usual $\prod_I F$ are all solid, which seems like a promising start. However, I have been unsuccessful in trying to show that one gets an abelian category.
If one tries to mimic the proof that solid abelian groups are an abelian category, the main obstruction we run into is $\operatorname{RHom}_F (\prod_I F, F)$. This is quite a difficult computation, and the main tool used to do this with condensed abelian groups is the short exact sequence $0 \to \mathbb{Z} \to \mathbb{R} \to \mathbb{R}/\mathbb{Z} \to 0$ so that we can use cohomology, but we don't have a similar sequence for $F$-modules.
The proof strategy for the analogue statement in $\mathbb{F}_p$-modules is heavily reliant on the fact that $\mathbb{F}_p$ is finite.
Essentially, the proof one needs is that $F_\blacksquare$ is an analytic ring. I know from Scholze's notes that it is suspected not all discrete rings form an analytic ring but I was hoping that this might be true for fields.
So my question is: should we expect a good theory of solid $F$-modules and why/why not?
EDIT: Some more details about my question. I am interested in whether $F_\blacksquare$ is analytic, where $F[S]^\blacksquare = \varprojlim F[S_i]$.
The motivation for this is a bit convoluted, but I'm in a situation where having the $\prod_I F$ as compact projective generators of an abelian category would be particularly convenient in order to extend certain functors only defined on these compact projective generators.