Let $K$ be a complete valued subfield of $\mathbb{C}_p$. Let $\mathcal O=\{z \in \mathbb C_p \colon \vert z \vert \leq 1\}$ be the ring of integers in $\mathbb C_p$ and $\mathfrak m=\{z \in \mathbb C_p \colon \vert z \vert < 1\}$ its unique maximal ideal. Let $R\subseteq K[[T]]$ denote the subring of power series that converge on $\mathfrak m$. Every $f \in R $ induces an analytic function $f \colon \mathfrak m \to \mathbb C_p$.
Suppose that the function induced by $f$ is bounded, i.e. there exists a real constant $C >0$ such that $\vert f(z) \vert \leq C$ for all $z \in \mathfrak m$. Are the coefficients of the power series $f$ then necessarily bounded?