If X is a curve over the p-adic rationals, then the l-adic etale Ceresa class of X is always trivial when l is different from p, for weight reasons. The p-adic Ceresa class, by contrast, contains much more information about X, and might reasonably be expected to be non-trivial for a suitably generic X. In this talk, I will describe some recently initiated work with Wanlin Li, where we show such a generic non-triviality result.