  We consider the singular numbers of a certain explicit continuous-time Markov
jump process on $\mathrm{GL}_N(\mathbb{Q}_p)$, which we argue gives the closest
$p$-adic analogue of multiplicative Dyson Brownian motion. We do so by
explicitly classifying the possible dynamics of singular numbers of processes
on $\mathrm{GL}_N(\mathbb{Q}_p)$ satisfying natural properties possessed by
Brownian motion on $\mathrm{GL}_N(\mathbb{C})$. Computing the evolution of
singular numbers explicitly, we find that the $N$-tuple of singular numbers in
decreasing order evolves as a Poisson jump process on $\mathbb{Z}^N$, with
ordering enforced by reflection off the walls of the positive type $A$ Weyl
chamber. This contrasts with -- and provides a $p$-adic analogue to -- the
behavior of classical Dyson Brownian motion, where ordering is enforced by
conditioning to avoid the Weyl chamber walls.
