We have investigated by angle-resolved photoemission spectroscopy changes induced by Xe adsorption on the spin-orbit-split bands of the epitaxial Ag(111) (root 3x root 3)R30 degrees-Bi surface alloy. We observe an extensive backfolding of the alloy band structure due to the formation of a 9 x 9 commensurate or quasicommensurate Xe overlayer and the opening of a small hybridization gap at the intersection of two sets of bands of (primarily) Bi sp(z) and p(xy) characters. The large spin-orbit splitting of the substrate bands is essentially unaffected by the Xe overlayer, in sharp contrast to the enhancement reported for the clean Au(111) surface. This points to different mechanisms at the origin of the splitting in the clean surfaces and in the surface alloy.