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 $D_K$? Can we write it as a power series algebra with 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?

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?

Let $K$ be a local field and $D_K$ the open unit ball, considered as a rigid space or adic space over $K$. What is the algebra of analytic functions on $D_K$? Can we write it as a power series algebra with some convergence condition?

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 $D_K$? Can we write it as a power series algebra with 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?