New models have recently been proposed in which a second Higgs doublet couples only to the lepton doublets and right-handed neutrinos, yielding Dirac neutrino masses. The vacuum value of this second "nu-Higgs" doublet is made very small by means of a very softly-broken Z 2 or U (1) symmetry. The latter is technically natural and avoids fine-tuning and very light scalars. We consider a supersymmetric version of this model, in which two additional doublets are added to the MSSM. If kinematically allowed, the decay of the heavy MSSM scalar into charged nu-Higgs scalars will yield dilepton events which can be separated from the W-pair background. In addition, the nu-Higgsinos can lead to very dramatic tetralepton, pentalepton and hexalepton events which have negligible background and can be detected at the LHC and the Tevatron. *