We describe the geometry of a top-dimensional quasiflat (that is, a quasi-isometrically embedded copy of euclidean space of maximal possible dimension) in a coarse median space of finite rank. We show that such a quasiflat is a bounded Hausdorff distance from a finite union of subsets, each of which has a simple structure. In particular, each of these subsets is the image of a direct product of real intervals or rays under a quasi-isometric embedding which preserves medians up to bounded distance. As one consequence, we recover the result of Behrstock, Hagen and Sisto, regarding quasiflats in asymphoric hierarchically hyperbolic spaces. In the case of a median metric space, one can strenghten the conclusion. In particular, we recover the result of Huang regarding quasiflats in CAT(0) cube complexes, namely that a top-dimensional quasiflat is a finite Hausdorff distance from a finite union of orthants. Examples of coarse median spaces include the mapping class groups of surfaces, as well as Teichmüller space in either the Teichmüller or the Weil-Petersson metric.
Preprint, Warwick, June 2019. Revised, December 2022.