In this talk we carry over some definitions and results from classical Deligne--Luzstig theory (for reductive groups over finite fields) to a setup over p-adic fields. We discuss p-adic Deligne--Lusztig spaces, defined as certain arc-sheaves on perfect algebras over the residue field. Then we look at some of their fundamental properties, as for example, (ind-)representability. In some cases, one can determine the cohomology of these spaces, and use it to construct smooth representations of p-adic reductive groups geometrically.