A non-archimedean $\lambda$-lemma

We provide a framework for studying the dynamics of families of one-variable rational functions parametrized by Berkovich spaces over a complete non-archimedean field. We prove a non-archimedean analogue of Ma~{n}'{e}, Sad, and Sullivan’s $\lambda$-Lemma and use this to show an equivalence of two stability conditions for families of rational functions parametrized by an open subset of the Berkovich affine line.