  Inspired by several alternative definitions of continued fraction expansions
for elements in $\mathbb Q_p$, we study $p$-adically convergent periodic
continued fractions with partial quotients in $\mathbb Z[1/p]$. To this end,
following a previous work by Brock, Elkies, and Jordan, we consider certain
algebraic varieties whose points represent formal periodic continued fractions
with period and preperiod of fixed lengths, satisfying a given quadratic
equation. We then focus on the $p$-adically convergent loci of these varieties,
characterizing the zero and one-dimensional cases.
