We give a generalisation of Deligne-Lusztig varieties for general and special
linear groups over finite quotients of the ring of integers in a
non-archimedean local field. Previously such a generalisation was given by
Lusztig by attaching certain varieties to unramified maximal tori inside Borel
subgroups. In this paper we associate a family of so-called extended
Deligne-Lusztig varieties to all tamely ramified maximal tori of the group.