  Special kinds of continued fractions have been proved to converge to
transcendental real numbers by means of the celebrated Subspace Theorem. In
this paper we study the analogous $p$--adic problem. More specifically, we deal
with Browkin $p$--adic continued fractions. First we give some new remarks
about the Browkin algorithm in terms of a $p$--adic Euclidean algorithm. Then,
we focus on the heights of some $p$--adic numbers having a periodic $p$--adic
continued fraction expansion and we obtain some upper bounds. Finally, we
exploit these results, together with $p$--adic Roth-like results, in order to
prove the transcendence of two families of $p$--adic continued fractions.
