  For $A \subseteq \mathbb{N}$, the question of when $R(A) = \{a/a' : a, a' \in
A\}$ is dense in the positive real numbers $\mathbb{R}_+$ has been examined by
many authors over the years. In contrast, the $p$-adic setting is largely
unexplored. We investigate conditions under which $R(A)$ is dense in the
$p$-adic numbers. Techniques from elementary, algebraic, and analytic number
theory are employed in this endeavor. We also pose many open questions that
should be of general interest.
