  We give a method for sampling points from an algebraic manifold (affine or
projective) over a local field with a prescribed probability distribution. In
the spirit of the previous work by Breiding and Marigliano on real algebraic
manifolds, our method is based on slicing the given variety with random linear
spaces of complementary dimension. We also provide an implementation of our
sampling method and discuss a few applications. As an application, we sample
from algebraic p-adic matrix groups and modular curves.
