Let $K$ be a local field and $D_K$ the open unit disk, considered as a rigid space or adic space over $K$. What is the algebra of analytic functions on $D_K$? Proposition 1.1 of this article describes functions on the punctured open disk as certain "Laurent series" (possibly with infinitely many terms of negative degree). Can we write functions on $D_K$ as power series satisfying some convergence condition?

A related question: suppose $K = k((t))$ for some field $k$. If I am not mistaken, $D_K = \text{Spa } k[[t]] \times \text{Spa } k((t))$ with the product taken in the category of adic spaces over $k$. So it seems to me that $D_K$ will have a "diagonal" $K$-point. Does this make sense?

anynon-archimedean field containing $k$ then ${\rm{Spa}}(k[\![t]\!],k[\![t]\!])\times_{{\rm{Spa}}(k,k)}{\rm{Spa}}(F,F^0)$ exists and is the (adic space associated to the) open unit disc over $F$, verified by universal property (exercise) using its "standard coordinate" (pulled back from the first factor) denoted as $t$ by abuse of notation. For $a \in F$ with $|a|<1$ there is a "classical point" cut out by the condition $t=a$. Now set $F=k(\!(t)\!)$ and $a=t\in F$ to get the diagonal point with abuse of notation. $\endgroup$