  Let X be a smooth, complex Fano 4-fold, and rho(X) its Picard number. If X
contains a prime divisor D with rho(X)-rho(D)>2, then either X is a product of
del Pezzo surfaces, or rho(X)=5 or 6. In this setting, we completely classify
the case where rho(X)=5; there are 6 families, among which one is new. We also
deduce the classification of Fano 4-folds with rho(X)>4 with an elementary
divisorial contraction sending a divisor to a curve.
