We study some possible astrophysical implications of a very weakly coupled ultralight dilaton-type scalar field. Such a field may develop an (approximately stable) network of domain walls. The domain wall thickness is assumed to be comparable with the thickness of the luminous part of the spiral galaxies. The walls provide trapping for galactic matter. This is used to motivate the very existence of the spiral galaxies. A zero-mode existing on the domain wall is a massless scalar particle confined to (1+2) dimensions. At distances much larger than the galaxy/wall thickness, the zero-mode exchange generates a logarithmic potential, acting as an additional term with respect to Newton's gravity. The logarithmic term naturally leads to constant rotational velocities at the periphery. We estimate the scalar field coupling to the matter energy–momentum tensor needed to fit the observable flat rotational curves of the spiral galaxies. The value of this coupling turns out to be reasonable — we find no contradiction with the existing data.