Homotopy theory allows us to study formal moduli problems via their tangent Lie algebras. We apply this general paradigm to Calabi-Yau varieties Z in characteristic p. First, we show that if Z has torsion-free crystalline cohomology and degenerating Hodge-de Rham spectral sequence (and for p=2 a lift to W/4), then its mixed characteristic deformations are unobstructed. This generalises the BTT theorem from characteristic 0 to characteristic p. If Z is ordinary, we show that it moreover admits a canonical (and algebraisable) lift to characteristic zero, thereby extending Serre-Tate theory from abelian varieties to Calabi-Yau varieties. This is joint work with Taelman, and generalises results of Achinger-Zdanowicz, Bogomolov-Tian-Todorov, Deligne-Nygaard, Ekedahl–Shepherd-Barron, Iacono-Manetti, Schröer, Serre-Tate, and Ward.