Extending the Minimal Supersymmetric Standard Model (MSSM) to explain small neutrino masses via the inverse seesaw mechanism can lead to a new light supersymmetric scalar partner which can play the role of inelastic dark matter (iDM). It is a linear combination of the superpartners of the neutral fermions in the theory (the light left-handed neutrino and two heavy Standard Model singlet neutrinos) which can be very light with mass in $∼ 5-20$ GeV range, as suggested by some current direct detection experiments. The iDM in this class of models has keV-scale mass splitting, which is intimately connected to the small Majorana masses of neutrinos. We predict the differential scattering rate and annual modulation of the iDM signal which can be testable at future Germanium- and Xenon-based detectors. ; Comment: 5 pages, 4 figures; final version (to appear in Phys. Rev. Lett.)