An almost reference (some assembly required) is N. Katz: Serre-Tate local moduli, Springer Lecture notes in Mathematics 868. By Lemma 1.1.2 an endomorphism lifts if and only if it lifts on the $p$-divisible group and for the canonical lift the endomorphism lifts trivially. I haven't looked at Drinfeld's original article to see if that is more suitable.

An almost reference (some assembly required) is N. Katz: Serre-Tate local moduli, Springer Lecture notes in Mathematics 868. By Lemma 1.1.2 an endomorphism lifts if and only if it lifts on the $p$-divisible group and for the canonical lift the endomorphism lifts trivially. I haven't looked at Drinfeld's original article to see if that is more suitable.

Addendum: There is no denying that my reference suffers in comparison with the two others given. I would like to point out however that Drinfeld's argument is very beautiful and arguably the slickest approach to canonical liftings (and Serre-Tate coordinates in general).