Vladimirov defined an operator on balls in $\mathbb Q_p$, the $p$-adic numbers, analogous to the Laplace operator in the real setting. Kochubei later gave a probabilistic interpretation of this operator. The Vladimirov--Kochubei operator generates a real-time diffusion process in the ring of $p$-adic integers, a Brownian motion in $\mathbb Z_p$. The current work proves that this process is a limit of discrete-time random walks. It motivates the construction of the Vladimirov--Kochubei operator, provides further intuition about ultrametric diffusion, and gives an example of the weak convergence of stochastic processes in a profinite group.