We study the distribution of singular numbers of products of certain classes of $p$-adic random matrices, as both the matrix size and number of products go to $\infty$ simultaneously. In this limit, we prove convergence of the local statistics to a new random point configuration on $\mathbb{Z}$, defined explicitly in terms of certain intricate mixed $q$-series/exponential sums. This object may be viewed as a nontrivial $p$-adic analogue of the interpolating distributions of Akemann-Burda-Kieburg arXiv:1809.05905, which generalize the sine and Airy kernels and govern limits of complex matrix products. Our proof uses new Macdonald process computations and holds for matrices with iid additive Haar entries, corners of Haar matrices from $\mathrm{GL}_N(\mathbb{Z}_p)$, and the $p$-adic analogue of Dyson Brownian motion studied in arXiv:2112.03725.