Let $\Gamma$ be a multiplicative group isomorphic to the $p$-adic integers $(\mathbb{Z}_p,+)$, and let $M$ be a discrete torsion $\Gamma$-module. Let $\gamma$ be a topological generator for $\Gamma$. Since $\Gamma$ is a topologically free group, we know that the cohomogical dimension of $\Gamma$ is $1$, hence $H^n(\Gamma, M)=0$ for every $n\ge2$. It also seems true that $$ M_\Gamma:=M/(\gamma-1)M\cong H^1(\Gamma,M), $$ but I can't find a proof of this, and I'm not sure that the conditions for $M$ are enough. The only hypothesis I may add is that $M^\Gamma$ is finite.
The most naural way to find an isomorphism seems to involve the morphism \begin{equation} \begin{split} &H^1(\Gamma,M)\longrightarrow M/(\gamma-1)M\\ &[\xi]\longmapsto [\xi(\gamma)] \end{split} \end{equation} where $[\cdot]$ denotes the class inside the quotient. This map is easily seen to be injective, but I can't prove the surjectivity.
I've found the solution in the book of Neukirch, Schmidt, Wingberg "Cohomology of Number Fields", Proposition 1.7.7. It says:
Let $G=\prod_{p\in S}\mathbb{Z}_p$ be a torsion-free procyclic group and let $A$ be a discrete $G$-module. If $A$ is $S$-torsion, then $H^1(G,A)\cong A_G$.
The proof involves a result in the cohomology of cyclic groups that is proved using cup products, but infact the map that gives the isomorphism is the one I mentioned.