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.