Median spaces provide a common framework for studying actions on real trees and CAT(0) cube complexes. In this talk, we focus on a particular class of infinite-rank median spaces, namely those for which every pair of points is separated by the interior of a half-space. Many interesting examples fall into this category, including the 0-skeletons of finite- or infinite-dimensional CAT(0) cube complexes endowed with the combinatorial metric, median spaces associated with real hyperbolic spaces, and locally convex median spaces. We will discuss several structural properties induced by this topological separation assumption and present super-rigidity results for actions of higher-rank lattices on such spaces.