  A theorem of Nomizu and van Est computes the cohomology of a compact
nilmanifold, or equivalently the group cohomology of an arithmetic subgroup of
a unipotent linear algebraic group over $\mathbb{Q}$. We prove a similar result
for the cohomology of a compact open subgroup of a unipotent linear algebraic
group over $\mathbb{Q}_{\ell}$ with coefficients in a complex of continuous
$\ell$-adic representations. We work with the triangulated categories defined
by Ekedahl which play the role of ``derived categories of continuous
$\ell$-adic representations''. This is motivated by Pink's formula computing
the derived direct image of an $\ell$-adic local system on a Shimura variety in
its minimal compactification, and its application to automorphic perverse
sheaves on Shimura varieties. The key technical result is the computation of
the cohomology with coefficients in a unipotent representation with torsion
coefficients by an explicit complex of polynomial cochains which is of finite
type.
