In a celebrated paper published in 1983, R. Mane, P. Sad, and D. Sullivan prove a result about holomorphic families of injections called the lambda-Lemma with impressive applications to the complex dynamics of families of one-variable rational functions. In this talk, I will discuss a framework for studying the dynamics of families of one-variable rational functions parametrized by Berkovich spaces over a complete non-archimedean field and a suitable non-archimedean analogue of the lambda-Lemma. I will explain how this can be used to prove the equivalence of two stability conditions in non-archimedean dynamics and also provide some interesting examples of non-archimedean families. Speaker(s): Thomas Silverman (Brown University)