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Let X be a smooth, complex Fano 4-fold, and rho(X) its Picard number. We showthat if rho(X)>12, then X is a product of del Pezzo surfaces. The proof relieson a careful study of divisorial elementary contractions f: X->Y such that theimage S of the exceptional divisor is a surface, together with the author'sprevious work on Fano 4-folds. In particular, given f: X->Y as above, undersuitable assumptions we show that S is a smooth del Pezzo surface with -K_Sgiven by the restriction of -K_Y.

Fano 4-folds with $b_2>12$ are products of surfaces