Let $k$ be an algebraically closed field of characteristic $p > 0$, $X$ a variety over $k$.
We say $X$ lifts to characteristic zero, if there exists a local ring $R$ containing $\mathbb Z$ with residue field $k$ and a flat scheme $\mathcal X$ over $R$, such that $$ \mathcal X \otimes_R k \simeq X.$$
In words: There exists a family over a ring of mixed characteristic, which has our $X$ for a special fibre.
For most classes of surfaces in Kodaira dimension zero, liftability is known: For K3 surfaces, liftability was established by Deligne, and for abelian surfaces, one can use more general theories developed for abelian varieties. Bi-elliptic and quasi-bi-elliptic surfaces can be dealt with explicitly.
As far as I know, there is nothing in the literature about Enriques Surfaces. For this class, the case p = 2 is the most interesting.
The question seems natural, so it would struck me as strange if it were still open. Does anyone around here know anything about this?
Thanks a lot.