I am looking for a reference to the following ``well known" fact.

Let $k$ be a perfect field of prime characteristic $p$ and $W(k)$ its ring of Witt vectors. Let $A_0$ be an ordinary abelian variety over $k$ and let $A$ be an abelian scheme over $W(k)$ that is the canonical (Serre--Tate) lifting of $A_0$. Then every endomorphism $u_0$ of $A_0$ lifts to an endomorphism of $A$. In other words, the natural map $End(A) \to End(A_0)$ is bijective.

My problem is that I need it for infinite $k$. (I know a couple of references that deal with finite $k$.)