  In this paper, we study a relation between digits of $p$-adic numbers and
Mahler's classification. We show that an irrational $p$-adic number whose
digits are automatic, primitive morphic, or Sturmian is an $S$-, $T$-, or
$U_1$-number in the sense of Mahler's classification. Furthermore, we give an
algebraic independence criterion for $p$-adic numbers whose digits are
Sturmian.
