  Let $\mathrm G / F$ be a reductive split $p$-adic group and let $\mathrm U$
be the unipotent radical of a Borel subgroup. We study the cohomology with
trivial $\mathbb Z_p$-coefficients of the profinite nilpotent group $N =
\mathrm U(\mathcal O_F)$ and its Lie algebra $\mathfrak n$, by extending a
classical result of Kostant to our integral $p$-adic setup. The techniques used
are a combination of results from group theory, algebraic groups and
homological algebra.
