On boundedness of periods of self maps of algebraic varieties

Let $X$ be an algebraic variety over a field $K\subset \overline{{\mathbb{Q}}_p}$ and $f$ be a self map. When $K$ is a local field, the boundedness of $f$-periods in $X(K)$ is a well studied question. We will study the same question for certain infinite extensions over ${\mathbb{Q}}_p$ under some conditions.