My question concerns the cohomology of a compact $p$-adic Lie group $G$ (wich is pro-$p$).

Let $M$ be a finite dimensional $\mathbb{Q}_p$-vector space with continuous linear $G$-action.

Lazard showed that, under certain hypothesis, the continuous cohomology of $G$ in $M$ is isomorphic to (a sub-module of) the cohomology of its Lie algebra in $M$.

My question is now : what if $M$ has coefficients in $\mathbb{F}_p$ ? Is there a way to compute the continuous cohomology of $G$ in $M$ using its Lie algebra ?

Thanks in advance.