  We study (smooth, complex) Fano 4-folds X having a rational contraction of
fiber type, that is, a rational map X-->Y that factors as a sequence of flips
followed by a contraction of fiber type. The existence of such a map is
equivalent to the existence of a non-zero, non-big movable divisor on X. Our
main result is that if Y is not P^1 or P^2, then the Picard number rho(X) of X
is at most 18, with equality only if X is a product of surfaces. We also show
that if a Fano 4-fold X has a dominant rational map X-->Z, regular and proper
on an open subset of X, with dim(Z)=3, then either X is a product of surfaces,
or rho(X) is at most 12. These results are part of a program to study Fano
4-folds with large Picard number via birational geometry.
