I don’t believe this is true; the following example comes from Debarres’s “Higher-dimensional algebraic geometry”. Take $C$ a curve of genus $g$, $d\geq g$, and let $C_d\to J^d(C)$ be the Abel—Jacobi map from the $d$-th symmetric product of $C$ to the Jacobian parametrizing degree-$d$ line bundles. The fiber of this map over $L$ is the projective space $\mathbb P H^0(C,L)$, so all fibers are Fano. If we let $[l]$ be the class of a line in some fiber, we have that the curves contracted by the map are precisely those numerically equivalent to $c[l]$, so that $ [l]$ is extremal, and $K_{C_d}.l=g-d-1<0$, so the Abel—Jacobi map is given by contracting a $K_{C_d}$-negative extremal ray, so this is a Fano contraction. Now take $g\geq 3$ and $d=g+1$; by Riemann—Roch a general fiber has dimension 1, while over any special line bundle (those with nontrivial first cohomology) the fiber will be larger; since for $g\geq 3$ Brill—Noether theory guarantees the existence of such special line bundles, we have that the fiber dimension jumps so the morphism is not flat.