Let $R$ be a ring that is $p$-adically complete for a prime $p$ and let $W(R)$ denote the ring of $p$-typical Witt vectors. Is it true that $W(R)$ is $p$-adically complete? (A ring $A$ is $p$-adically complete if the map $A \rightarrow \varprojlim(A/p^nA)$ is bijective.) A reference that contains a proof or a counterexample would suffice.

I am familiar with the case when $R$ is a perfect $\mathbb{F}_p$-algebra and am interested in the general case.