A variety that cannot be dominated by one that lifts

We prove a precise version of a theorem of Siu and Beauville on morphisms to higher genus curves, and use it to show that if a variety $X$ in characteristic $p$ lifts to characteristic $0$, then any morphism $X\to C$ to a curve of genus $g\ge 2$ can be lifted along. We use this to construct, for every prime $p$, a smooth projective surface $X$ over ${\bar{\mathbb{F}}}_p$ that cannot be rationally dominated by a smooth proper variety $Y$ that lifts to characteristic $0$.