The discovery of ($4\times 4$) silicene formation on Ag(111) raised the question whether silicene maintains its Dirac fermion character, similar to graphene, on a supporting substrate. Previous photoemission studies indicated that the π band forms Dirac cones near the Fermi energy, while theoretical investigations found it shifted at deeper binding energy. By means of angle-resolved photoemission spectroscopy and density-functional theory calculations we show instead that the π-symmetry states lose their local character and the Dirac cone fades out. The formation of an interface state of free-electron-like Ag origin is found to account for spectral features that were theoretically and experimentally attributed to silicene bands of π character.