Let $\mathbb{Z}_p$ be te ring of p-adic integers and let $T$,$T'$ be two free $\mathbb{Z}_p$-module with a continuous action of $G_{\mathbb{Q}_p}$ (the absolute Galois group of $\mathbb{Q}_p$).
It is not hard to see that if $T$ and T' are of rank one, then there is a non zero $G_{\mathbb{Q}_p}$-equivariant map T \to T' \otimes \mathbb{Q}_p / \mathbb{Z}_p if and only if T \simeq T' mod $p$ (just write down explicitely the characters).
Is this true in a more general case, namely when T' and T' mod $p$ are irreducible and $T$ is any $\mathbb{Z}_p$-representation ?