On the Exceptional Sets of $p$-adic Transcendental Analytic Functions

In this paper, we study the exceptional sets $S_f$ of $p$-adic transcendental analytic functions $f$ with rational and algebraic coefficients. We establish a necessary condition for a subset $S \subseteq \overline{\mathbb{Q}} \cap B(0, \rho)$ to be the exceptional set of a $p$-adic transcendental analytic function with rational coefficients, demonstrating that, in general, the answer to Mahler's Problem C over $\mathbb{C}_p$ is negative. However, we prove that if $S$ is closed under algebraic conjugation and contains 0, there exist uncountably many transcendental analytic functions $f \in \mathbb{Q}_{\rho}[[z]]$ such that $S_f = S$. Furthermore, if $\rho \geq 1$, $f$ can be taken in $\mathbb{Z}_{\rho}[[z]]$. Additionally, we demonstrate that any $S \subseteq \overline{\mathbb{Q}} \cap B(0, \rho)$ containing 0 can be the exceptional set of uncountably many transcendental analytic functions $f \in \overline{\mathbb{Q}}_{\rho}[[z]]$.