The OPERA collaboration has claimed the discovery of supeluminal neutrino propagation. However the superluminal interpretation of the OPERA result was refuted by Cohen and Glashow because it was shown that such superluminal neutrinos would lose energy rapidly via the bremsstrahlung of electron-positron pairs (arXiv:1109.6562). We note that the superluminal interpretation is still possible if there exists a new (dark) neutrino which can propagate with a superluminal velocity and which couples with usual neutrinos only via the mass mixing leading to neutrino oscillations. It is supposed that the physical laws are invariant under rotations and translations in a preferred reference frame. Two possible pictures to parameterize departures from Lorentz invariance are discussed: a "conventional" tachyonic dark neutrino and the modification of the Lagrangian by adding perturbations with the maximum attainable speed of the dark neutrino which is larger than the speed of light in vacuum. We analyze also the MINOS and SN1987a data and show that they are consistent with the conjecture that there exists the superluminal dark neutrino.