A dense subgroup with completion not isomorphic to the big (pro-p) group?

Question

This is an (early) exercise from the book "Analytic Pro-p groups": (p.31, ex. 3(iii))

Give an example of a finitely generated pro-$p$ group $G$ and a dense subgroup $H$ of $G$, with $H$ finitely generated as an abstract group , such that $\hat{H_p} \ncong G$.

Any ideas would be really appreciated. I do suspect that the answer would be a linear group though.