  We study combinatorial properties of convex sets over arbitrary valued
fields. We demonstrate analogs of some classical results for convex sets over
the reals (e.g. the fractional Helly theorem and B\'ar\'any's theorem on points
in many simplices), along with some additional properties not satisfied by
convex sets over the reals, including finite breadth and VC-dimension. These
results are deduced from a simple combinatorial description of modules over the
valuation ring in a spherically complete valued field.
