  The classical theory of continued fractions has been widely studied for
centuries for its important properties of good approximation, and more recently
it has been generalized to $p$-adic numbers where it presents many differences
with respect to the real case. In this paper we investigate periodicity for the
$p$-adic continued fractions introduced by Browkin. We give some necessary and
sufficient conditions for periodicity in general, although a full
characterization of $p$-adic numbers having purely periodic Browkin continued
fraction expansion is still missing. In the second part of the paper, we
describe a general procedure to construct square roots of integers having
periodic Browkin $p$-adic continued fraction expansion of prescribed even
period length. As a consequence, we prove that, for every $n \ge 1$, there
exist infinitely many $\sqrt{m}\in \QQ_p$ with periodic Browkin expansion of
period $2^n$, extending a previous result of Bedocchi obtained for $n=1$.
