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Let $E$ be a complete Hausdorff locally convex space over $\mathbb{C}_{p},$let $A\in\mathcal{L}(E)$ such that $(I-\lambda A)^{-1}$ is analytic on itsdomain. In this paper, we give a necessary and sufficient condition on theresolvent of $A$ such that $(A^{n})_{n\in\mathbb{N}}$ is equi-continuous.

A Hille-Yoshida-Phillips theorem for discrete semigroups on complete ultrametric locally convex spaces