I will discuss joint work with Lukas Brantner, in which we use derived algebraic geometry to study deformations of varieties with trivial canonical bundle in characteristic p. We prove a positive characteristic analogue of the Bogomolov-Tian-Todorov theorem (which states that deformations of varieties with trivial canonical bundle are unobstructed in characteristic zero), and show that 'ordinary' varieties with trivial canonical bundle admit canonical lifts to characteristic zero (generalizing earlier results of Serre-Tate for abelian varieties, and Deligne and Nygaard for K3 surfaces).