We investigate the existence of the pure spin ratchet effect in a dissipative quasi-one-dimensional system with Rashba spin-orbit interaction. The system is additionally placed into a transverse uniform stationary in-plane magnetic field. It is shown that the effect exists at low temperatures and pure spin currents can be generated by applying an unbiased ac driving to the system. An analytical expression for the ratchet spin current is derived. From this expression it follows that the spin ratchet effect appears as a result of the simultaneous presence of the spin-orbit interaction, coupling between the orbital degrees of freedom and spatial asymmetry. In this paper we consider the case of a broken spatial symmetry by virtue of asymmetric periodic potentials. It turns out that an external magnetic field does not have any impact on the existence of the spin ratchet effect but it influences its efficiency, enhancing or reducing the magnitude of the spin current.