Local ring of an affinoid K-space $Spec A$ is not $A_m$

Question

Let $X$ be an affinoid K-space. That is $X=Spec A$ where $A$ is a quotient of the Tate algebra. In Bosch 'lectures on Formal and Rigid Geometry' we find the curious statement that the canonical map $O_X \to O_{X,x}$ factors as $O_X \to A_m \to O_{X,x}$ where $m$ is the maximal ideal corresponding to $x\in X$, implying that the local ring $O_{X,x}$ can be larger than $A_m$. Is this true? Is there an explicit example?