A 2D nonlinear lattice model has been suggested for studies of proton dynamics in ice-like networks. The model does not violate the fundamental Bernal-Fowler rules and therefore it adequately describes dichotomously branching transfers of protons in 2D and 3D hydrogen-bonded networks. Nonlinear collective excitations such as 2D topological kinks have been shown to describe positive and negative ionic defects in ice. The dynamical process of creation of ionic defect pairs and the dependence of its density on temperature has been studied in the frame of this model. The soliton dynamics of the 2D topological defects is shown to differ essentially from the well-known soliton motion in 1D kink-bearing systems. Thus, the defects cannot propagate so freely as usual soliton-like quasi- particles because of the dichotomy of proton transfers in hydrogen bonds.