We report on the first successful growth of wurtzite (WZ) GaBiAs nanowires (NWs) and reveal the effects of Bi incorporation on the electronic band structure by using polarization-resolved optical spectroscopies performed on individual NWs. Experimental evidence of a decrease in the band-gap energy and an upward shift of the topmost three valence subbands upon the incorporation of Bi atoms is provided, whereas the symmetry and ordering of the valence band states remain unchanged, that is, Γ9, Γ7, and Γ7 within the current range of Bi compositions. The extraordinary valence band structure of WZ GaBiAs NWs is explained by anisotropic hybridization and anticrossing between p-like Bi states and the extended valence band states of host WZ GaAs. Moreover, the incorporation of Bi into GaAs is found to significantly reduce the temperature sensitivity of the band-gap energy in WZ GaBiAs NWs. Our work therefore demonstrates that utilizing dilute bismide alloys provides new avenues for band-gap engineering and thus photonic engineering with NWs.