A question about complete Noetherian local $\mathbb{Z}_p$-algebras

Question

Let $A=\mathbb{Z}_p[[X_{1},\cdots,X_{r}]]/(f_{1},\cdots,f_{s})$ be a complete Noetherian local $\mathbb{Z}_p$-algebra where $\mathbb{Z}_p$ is the ring of $p$-adic integers. Suppose that the quotient ring $A/(p)$ is finite. Is it true that $A$ is a finitely generated module over $\mathbb{Z}_p$? I know that it is true if $r\geq s$. What about $r<s$? By Lemma, it suffices to show that $A$ is separated for the $p$-adic topology.