  Let $p>2$ be a prime number. In this short note, we calculate explicitly the
unitary dual and the matrix coefficients of the Heisenberg group over the
$p$-adic integers. As an application, we consider directional
Vladimirov-Taibleson derivatives, and some polynomials in these operators. In
particular, we calculate explicitly the spectrum of the Vladimirov
sub-Laplacian, and show how it provides a non-trivial example of a sub-elliptic
operator on compact graded $p$-adic Lie groups.
