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In this paper we investigate $p$-adic self-similar sets and $p$-adicself-similar measures. We show that $p$-adic self-similar sets are $p$-adicpath set fractals, and that the converse is not necessarily true. For $p$-adicself-similar sets and $p$-adic self-similar measures, we show the existence ofa unique essential class. We show that, under mild assumptions, the decimationof $p$-adic self-similar sets is maximal. For $p$-adic self-similar measures,we show that many results involving local dimension are similar to those oftheir real counterparts, with fewer complications. Most of these results usethe additional structure of self-similarity, and are not true in general for$p$-adic path set fractals.

Self-similar sets and self-similar measures in the $p$-adics