  We show that singular numbers (also known as invariant factors or Smith
normal forms) of products and corners of random matrices over $\mathbb{Q}_p$
are governed by the Hall-Littlewood polynomials, in a structurally identical
manner to the classical relations between singular values of complex random
matrices and Heckman-Opdam hypergeometric functions. This implies that the
singular numbers of a product of corners of Haar-distributed elements of
$\text{GL}_N(\mathbb{Z}_p)$ form a discrete-time Markov chain distributed as a
Hall-Littlewood process, with the number of matrices in the product playing the
role of time. We give an exact sampling algorithm for the Hall-Littlewood
processes which arise by relating them to an interacting particle system
similar to PushTASEP. By analyzing the asymptotic behavior of this particle
system, we show that the singular numbers of such products obey a law of large
numbers and their fluctuations converge dynamically to independent Brownian
motions. In the limit of large matrix size, we also show that the analogues of
the Lyapunov exponents for matrix products have universal limits within this
class of $\text{GL}_N(\mathbb{Z}_p)$ corners.
