Let $A$ be a Huber (i.e. f-adic) ring, meaning a topological ring with an open subring $A_0$ which is adic and has finitely generated ideal of definition.

Is there a good notion of closed disk of radius $r > 0$ over $\text{Spa } A$? If $A$ is a nonarchimedean field this is $$\text{Spa } \{ \sum_{n \geq 0} a_nx^n \ | \ |a_n|r^n \to 0 \}.$$ I'm hoping for a nice version of this in families, but I don't think we should just impose the condition above for all continuous valuations on $A$: it seems like nilpotents would cause problems. Ideally I would want functions on a disk of radius $0 < r < 1$ over any discrete ring $A$ to be $A[[x]]$.

And what about annuli? Again it would be nice if there was a uniform definition so that functions on an annulus with radii $0 \leq r_1 < r_2 < 1$ over $A$ discrete are $A((x))$.