  We give a stochastic proof of the finite approximability of a class of
Schr\"odinger operators over a local field, thereby completing a program of
establishing in a non-Archimedean setting corresponding results and methods
from the Archimedean (real) setting. A key ingredient of our proof is to show
that Brownian motion over a local field can be obtained as a limit of random
walks over finite grids. Also, we prove a Feynman-Kac formula for the finite
systems, and show that the propagator at the finite level converges to the
propagator at the infinite level.
