We study the localized states of an extra electron in an anisotropic quasi-two-dimensional system in which the electron-lattice interaction and the anharmonicity of the lattice vibrations are dominant in one direction. This model describes layers of polydiacetylene or other polymer chains, beta sheets of polypeptides, multilevel microstructures of conjugated polymers, and other low-dimensional systems. It is shown that for appropriate parameter values of the system an extra electron can excite a soliton-like mobile wave of the lattice deformation, within which it can get self-trapped. Such a bound state of an electron and the lattice deformation form a nonlinear two-component polaron-like entity, which can propagate with minimum of the energy dissipation. Our findings are based on the variational approach and the full numerical solution of the coupled system of nonlinear equations. These results suggest that the experimentally measured charge and energy transport over macroscopic distances in the above-mentioned systems can be provided by the soliton mechanism and thus have a potential impact on the theoretical background of the numerous applications of low-dimensional materials in nanoelectronics.