We study dynamics of rational maps of degree at least 2 with coefficients in the field $\mathbb{C}_p$, where $p > 1$ is a fixed prime number. The main ingredient is to consider the action of rational maps in $p$-adic hyperbolic space, denoted $\mathbb{H}_p$. Hyperbolic space $\mathbb{H}_p$ is provided with a natural distance, for which it is connected and one dimensional (an $\mathbb{R}$-tree). This advantages with respect to $\mathbb{C}_p$ give new insight into dynamics; in this paper we prove the following results about periodic points. In forthcoming papers we give applications to the Fatou/Julia theory over $\mathbb{C}_p$.

First we prove that the existence of two non-repelling periodic points implies the existence of infinitely many of them. This is in contrast with the complex setting where there can be at most finitely many non-repelling periodic points. On the other hand we prove that every rational map has a repelling fixed point, either in the projective line or in hyperbolic space.

We also caracterise those rational maps with finitely many periodic points in hyperbolic space. Such a rational map can have at most one periodic point (which is then fixed) and we characterise those rational maps having no periodic points and those rational maps having precisely one periodic point in hyperbolic space.

We also prove a formula relating different objects in the projective line and in hyperbolic space, which are fixed by a given rational map. Finally we relate hyperbolic space in the form given here, to well known objects: the Bruhat-Tits building of $PSL(2, \mathbb{C}_p)$ and the Berkovich space of $\mathbb{P}(\mathbb{C}_p)$.