Given a prime $p$, let $P(t)$ be a non-constant monic polynomial in $t$ over the ring $\mathbb{Z}_{p}$ of $p$-adic integers. Let $X_n$ be the $n \times n$ uniformly random (0,1)-matrix over $\mathbb{Z}_{p}$. We compute the asymptotic distribution of the cokernel of $P(X_n)$ as $n$ goes to infinity. When $P(t)$ is square-free modulo $p$, this lets us compute the asymptotic distribution of the Smith normal form of $P(X_n)$. In fact, we shall consider the same problem with a more general random matrix $X_n$, which also includes the example of a Haar-random matrix. Our work crucially uses a recent work of W. Sawin and M. M. Wood which shows that the moments of finite size modules over any ring determine their distribution. This is joint work with Myungjun Yu. https://arxiv.org/abs/2303.09125