  In this article, we develop a dynamical theory for what shall be called a
skew product on the Berkovich projective line, $\phi_*:
\mathbb{P}^1_{\text{an}}(K) \to \mathbb{P}^1_{\text{an}}(K)$ over a
non-Archimedean field $K$. These functions are defined algebraically yet
strictly generalise the notion of a rational map on $\mathbb{P}^1_{\text{an}}$.
We describe the analytical, algebraic, and dynamical properties of skew
products, including a study of periodic points, and a Fatou/Julia dichotomy.
The article culminates with the classification of the connected components of
the Fatou set.
