If $M$ is a p-adically complete abelian group, so that it's a $\mathbb{Z}_p$-module, and we have an injective homomorphism $\phi: \mathbb{Z} \hookrightarrow M$, is it true that the induced homomorphism $\hat{\phi}: \mathbb{Z}_p \rightarrow M $ is also injective? If this is not true in general, would it be true if $\phi$ was also continuous with respect to the p-adic topologies?
This kind of statement feels too good to be true, and I'm sure there is a super basic counterexample somewhere out there, but I just can't come up with it. Any help will be greatly appreciated!
HINT: The nonzero ideals of $\Bbb{Z}_p$ are of the form $p^n\Bbb{Z}_p$, so if the induced map is not injective then $\hat{\phi}(p^n)=0$ for some $n\in\Bbb{N}$.