  In 1976, Deligne and Lusztig realized the representation theory of finite
groups of Lie type inside \'etale cohomology of certain algebraic varieties.
Recently, a $p$-adic version of this theory started to emerge: there are
$p$-adic Deligne-Lusztig spaces, whose cohomology encodes representation
theoretic information for $p$-adic groups - for instance, it partially realizes
local Langlands correspondence with characteristic zero coefficients. However,
the parallel case of coefficients of positive characteristic $\ell \neq p$ has
not been inspected so far. The purpose of this article is to initiate such an
inspection. In particular, we relate cohomology of certain $p$-adic
Deligne-Lusztig spaces to Vign\'eras's modular local Langlands correspondence
for $\mathbf{GL}_n$.
