Aqueous solutions of lithium chloride are an excellent model system for studying the dynamics of water molecules down to low temperatures without freezing. The apparent dynamic crossover observed in an aqueous solution of LiCl at about 220 to 225 K [Mamontov, JPCB 2009, 113, 14073] is located practically at the same temperature as the crossover found for pure water confined in small hydrophilic pores. This finding suggests a strong similarity of water behavior in these two types of systems. At the same time, studies of solutions allow more effective explorations of the long-range diffusion dynamics, because the water molecules are not confined inside an impenetrable matrix. In contrast to the earlier incoherent quasielastic neutron scattering results obtained for the scattering momentum transfers of 0.3 angstrom(-1) <= Q <= 0.9 angstrom(-1), our present incoherent neutron spin-echo measurements at a lower Q of 0.1 angstrom(-1) exhibit no apparent crossover in the relaxation times down to 200 K. At the same time, our present nuclear magnetic resonance measurements of the diffusion coefficients clearly show a deviation at the lower temperatures from the non-Arrhenius law obtained at the higher temperatures. Our results are consistent with a scenario in which more than one relaxational component may exist below the temperature of the dynamic crossover in water.