Let $K$ be a complete non-archimedean field and let $X$ be an affinoid $K$-space. Then the structure sheaf $\mathcal O_X$ of $X$ is defined first on the "weak Grothendieck topology", ie on affinoid subdomains, and then by extending the sheaf to the finer "strong Grothendieck topology" $\mathcal T$, also called Tate topology.
Now while on affinoid subdomains $U\subset X$, the structure sheaf takes values $\mathcal O_X(U)$ in affinoid $K$-algebras, this is not necessarily true any more for general admissible opens, because for instance the value category needs to have arbitrary Cartesian products. In most of the literature I have seen, the structure sheaf is therefore considered as a functor
$$\mathcal O_X: \text{Cat} \mathcal T \longrightarrow K{-}Alg$$
from the underlying category of the Grothendieck topology to just $K$-algebras. Consequently, rigid spaces are later defined to be certain spaces carrying sheaves of locally ringed $K$-algebras. Although I can believe that this works perfectly well, I somehow feel that in doing so we forget some of the structure of $\mathcal O_X(U)$ even for general admissible opens. My question is therefore:
What category does $\mathcal O_X$ naturally take values in? In other words, does $\mathcal O_X$ factor through some forgetful functor to $K{-}Alg$?
The reason I am asking is that for the examples I first thought of, I always seem to get at least Banach $K$-algebras, although not necessarily topologically of finite type over $K$.
If you just read until here and you think you might know an answer to the question, I would be very happy to get an answer from you, you don't need to read any further. If on the other hand you think that this is an unreasonable question, let me add a few more words about why I think we might get more structure:
The way that $\mathcal O_X$ is extended to the finer Grothendieck topology is by a Lemma about sheaves on Grothendieck topologies, so my rigid geometry books don't give all the technicalities. But I think the way this works is by defining $\mathcal O_X(U)$ for an admissible open $U\subset X$ to be the limit $$\varinjlim_{\mathfrak U} H^0(\mathfrak U,\mathcal O_X)$$ where $\mathfrak U = (U_i)_{i \in I}$ ranges over all admissible coverings by affinoid open subsets of $X$, partially ordered by refinement, and where $$ H^0(\mathfrak U,\mathcal O_X) = \ker \left( \prod_{i \in I} \mathcal O_X(U_i) \rightrightarrows \prod_{i,j \in I} \mathcal O_X(U_i\cap U_j) \right).$$
So $K$-algebras is a sensible choice for the value category because it has both direct limits and Cartesian products. Banach $K$-algebras doesn't have direct limits (I think), but since $ H^0(\mathfrak U,\mathcal O_X)$ is the kernel of a continuous map from a product of Banach $K$-algebras, it is a Banach $K$-algebra at least in some cases. So maybe we are only taking "lucky limits" for which some of the topological structure is preserved?
Let me give an example: Let $A = K\langle \zeta\rangle$, $X = Sp(A)$ the unit disc and consider the set $$D(\zeta) = \{x \in X \mid |\zeta(x)|>0\}$$ ie the unit disc pointed at the origin. Then if I am not mistaken, the functions on this subspace are $$\mathcal O_X(D(\zeta)) = \left\{\sum_{n\in \mathbb Z}a_n\zeta^n \middle | |a_n|\to 0 \text{ for } n\to \infty\text{, and }x^n|a_{-n}|\to 0 \text{ for } n\to \infty\text{ and any }x\in \mathbb R_{>0}\right\}$$ which is indeed complete wrt the natural extension of the Gauss-norm. Here I think that we may be taking advantage of the fact that a neighborhood basis of the origin is given by Weierstraß-domains, and that we can therefore always find a refining covering of the form $\bigcup_{n\in \mathbb N}X(\zeta^{-1};\epsilon^n)$ for some $\epsilon>0$. In particular, $\mathcal O_X(D(\zeta))$ is an inverse limit of Banach-$K$-algebras and therefore again a Banach-$K$-algebra. This makes me wonder:
Does the answer to the above question change if we restrict $\mathcal O_X$ to the Zariski topology?