Abstract. Radio observations show that magnetic fields are present in dwarf irregular galaxies (dIrr) and its strength is comparable to that found in spiral galaxies. Slow rotation, weak shear and shallow gravitational potential are the main features of a typical dIrr galaxy. These conditions of the interstellar medium in a dIrr galaxy seem to unfavourable for amplification of the magnetic field through the dynamo process. Cosmic-ray driven dynamo is one of the galactic dynamo model, which has been successfully tested in case of the spiral galaxies. We investigate this dynamo model in the ISM of a dIrr galaxy. We study its efficiency under the influence of slow rotation, weak shear and shallow gravitational potential. Additionally, the exploding supernovae are parametrised by the frequency of star formation and its modulation, to reproduce bursts and quiescent phases. We found that even slow galactic rotation with a low shearing rate amplifies the magnetic field, and that rapid rotation with a low value of the shear enhances the efficiency of the dynamo. Our simulations have shown that a high amount of magnetic energy leaves the simulation box becoming an efficient source of intergalactic magnetic fields.