Abstract We define a stratification of Deligne–Lusztig varieties and their parahoric analogues which we call the Drinfeld stratification. In the setting of inner forms of $${{\,\mathrm{GL}\,}}_n$$ GL n , we study the cohomology of these strata and give a complete description of the unique closed stratum. We state precise conjectures on the representation-theoretic behavior of the stratification. We expect this stratification to play a central role in the investigation of geometric constructions of representations of p-adic groups.