  We study a non-archimedean (NA) version of transportation problems and
introduce naturally arising ultra-norms which we call Kantorovich ultra-norms.
For every ultra-metric space and every NA valued field (e.g., the field
$\mathbb Q_{p}$ of $p$-adic numbers) the naturally defined inf-max cost formula
achieves its infimum. We also present NA versions of the Arens-Eells
construction and of the integer value property. We introduce and study free NA
locally convex spaces. In particular, we provide conditions under which these
spaces are normable by Kantorovich ultra-norms and also conditions which yield
NA versions of Tkachenko-Uspenskij theorem about free abelian topological
groups.
