Assume $p\neq q$ are two primes. By $\mathbb Z_q$ I mean the p-adic integers, and by Hom I mean group homomorphisms.
If $\rm{Hom}(\mathbb Z_q, \mathbb C_p) \neq 0$: is the set of continuous homomorphisms 0?
If that's too difficult: How about $\rm{Hom}(\mathbb Z_q, \mathbb A_p)$ with $\mathbb A_p$ the ring of integers in $\mathbb C_p$?