We investigate theoretically the influence of the spin-orbit interaction of Rashba type on the magnetoresistance of a semiconducting ferromagnetic nanostructure with a laterally constrained domain wall. The domain wall is assumed sharp (on the scale of the Fermi wavelength of the charge carriers). It is shown that the magnetoresistance in such a case can be considerably large, which is in qualitative agreement with recent experimental observations. It is also shown that spin-orbit interaction may result in an increase of the magnetoresistance. The role of localization corrections is also briefly discussed.