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

Number Theory;Classical Analysis and ODEs