  Let $X$ be a smooth projective variety defined over an algebraically closed
field of positive characteristic $p$ whose tangent bundle is nef. We prove that
$X$ admits a smooth morphism $X \to M$ such that the fibers are Fano varieties
with nef tangent bundle and $T_M$ is numerically flat. We also prove that
extremal contractions exist as smooth morphisms.
  As an application, we prove that, if the Frobenius morphism can be lifted
modulo $p^2$, then $X$ admits, up to a finite \'etale Galois cover, a smooth
morphism onto an ordinary abelian variety whose fibers are products of
projective spaces.
