The height of an algebraic number $\alpha$ is a measure of how arithmetically
complicated $\alpha$ is. We say $\alpha$ is totally $p$-adic if the minimal
polynomial of $\alpha$ splits completely over the field $\mathbb{Q}_p$ of
$p$-adic numbers. In this paper, we investigate what can be said about the
smallest nonzero height of a degree $3$ totally $p$-adic number. In particular,
we provide an algorithm to determine, given a prime $p$, the smallest height of
a degree $3$ totally $p$-adic number.