A new scenario for creation of galactic magnetic fields is proposed which is operative at the cosmological epoch of the galaxy formation, and which relies on unconventional properties of dark matter. Namely, it requires existence of feeble but long range interaction between the dark matter particles and electrons. In particular, millicharged dark matter particles or mirror particles with the photon kinetic mixing to the usual photon can be considered. We show that in rotating protogalaxies circular electric currents can be generated by the interactions of free electrons with dark matter particles in the halo, while the impact of such interactions on galactic protons is considerably weaker. The induced currents may be strong enough to create the observed magnetic fields on the galaxy scales with the help of moderate dynamo amplification. In addition, the angular momentum transfer from the rotating gas to dark matter component could change the dark matter profile and formation of cusps at galactic centers would be inhibited. The global motion of the ionized gas could produce sufficiently large magnetic fields also in filaments and galaxy clusters.The origin of large-scale magnetic fields remains one of very deep cosmological mysteries. Magnetic fields are detected in galaxies of all types and they constitute a very important component of the galactic dynamics since they are relevant for compression of the gas clouds, influence star formation process, and determine the spectrum of the galactic cosmic rays. The Milky Way, for example, possesses the magnetic field of a few µG over the plane of its disc, with a coherence length of a few kpc. Similar magnetic fields have been a