p-adic character varieties are rigid analytic moduli spaces of p-adic representations of fundamental groups. For curves, I will explain how these varieties lead to a non-abelian generalisation of the p-adic Hodge-Tate decomposition, by way of an arithmetic Hitchin fibration. I will then report on joint work in progress with Magnus Carlson and Ruth Wild which uses this perspective for a new approach to Mochizuki's Theorem that hyperbolic curves are anabelian.