Let $(k,|.|)$ be a complete non-archimedean valued field. Let $D$ be the open unit disc over $k$. (Anything I write could be adapted to the case of an open annulus.) The ring $\mathcal{O}(D)$ of analytic functions on $D$ admits an explicit description: it consists of the series $\sum_{n\ge 0} a_n T^n$ such that for any $r<1$, the sequence $(|a_n|r^n)_{n\ge 0}$ tends to 0. If such a function is bounded, its uniform norm is precisely $\sup_{n\ge 0}(|a_n|)$.

For some purposes, it is convenient to consider smaller rings: the ring of bounded functions or the ring of analytic elements, for instance. Let me recall that this latter ring $\mathcal{H}(D)$ is defined as the completion of $k(T) \cap \mathcal{O}(D)$ (*i.e.* rational functions with no poles in $D$) for the uniform norm on the disc. Remark that this definition depends on the choice of a coordinate $T$ on $D$. Those functions have nice properties: for example, they only have a finite number of zeros on $D$. We refer to Gilles Christol's book (chapter I) for a more detailed account.

Consider the disc $D$ inside the affine line $\mathbf{A}^1$. Let $X$ be an affinoid domain of $\mathbf{A}^1$ that contains $D$. One may check that any function on $X$ restricts to an analytic element on $D$, whatever the choice of the coordinate (because the approximation property actually holds for any function on $X$).

The question is the following: if $D$ is embedded inside an arbitrary affinoid space $X$, do functions on $X$ still restrict to analytic elements?

The issue is that the choice of a coordinate in the definition of $\mathcal{H}(D)$ makes it difficult to use outside the affine line. I would appreciate any idea that helps recognizing analytic elements on a disc inside an arbitrary curve.