  In characteristic $0$, the Campana-Peternell conjecture claims that the only
smooth Fano variety with nef tangent bundle should be homogeneous. In this
paper, we study the positive characteristic version of the Campana-Peternell
conjecture. In particular, we give an affirmative answer for Fano $4$-folds
with nef tangent bundle and Picard number greater than one.
