$p$-adic Brownian motion

We define $p$-adic Brownian motion (Wiener process) and study its properties. We construct a presentation of the trajectories of this process by their series expansions with respect to van der Put's basis and show that they are nowhere differentiable functions satisfying the $p$-adic Lipschitz condition of order $1$. We define the $p$-adic Wiener measure on the space of continuous functions and study its properties.