Basing on Dirac equation for interacting massless fermions, we propose a nonlinear model that describes a possible mechanism of ferromagnetism in graphene structures, resulting from electron-electron interaction and spontaneous breaking of spin symmetry of valence electrons. Qualitative predictions of the model are important for practical applications in spintronics. Localized kink-antikink patterns of valence electron spin density on the graphene surface are calculated, their interaction is described, and, finally, the formation of their quasi-bound metastable states (breathers) is investigated. The spectrum of breathers is calculated in both the analytical and the numerical form. Possible inverted population of the appropriate states may be used to generate quantum coherent nonlinear spin waves that can find practical applications in nanoelectronics and spintronics.