Infinite product in p-adic fields

Question

I have a (probably simple) question about the convergence of an infinite product in p-adic fields. In particular we consider the completion $\mathbb{C}_p$ of an algebraic closure of the p-adic numbers $\mathbb{Q}_p$. Then we have given two sequences $(a_i)_i$ and $(b_i)_i$ in $\mathbb{C}_p^\times$ , where $|a_i|=1$, $|b_i|<1$ and the limit $\prod_{i=0}^\infty a_i$ exists. Is it then true, that the infinite product $\prod_{i=0}^\infty (a_i+p^ib_i)$ converges with limit $\prod_{i=0}^\infty a_i$ too?
Thanks for any comment :)