The puzzling electronic and transport properties of the Ga(AsBi) alloy are investigated for a wide range of Bi‐concentrations (x = 0–10.6%) by means of various experimental techniques in high magnetic fields (B up to 30 T): magneto‐photoluminescence spectroscopy, magneto‐far‐infrared (FIR) absorption spectroscopy, and Hall effect measurements. Our experimental findings suggest that the strength of hybridization of the continuum states of the valence and conduction bands with the Bi‐related electronic levels depends on the Bi‐concentration, thus leading to band edges with a localized (for x < 6%) or band‐like character (for x > 8%). We report an unusual compositional‐dependence of the exciton reduced mass (µexc), whose value can be larger (for x < 6%) or smaller (for x > 8%) than in GaAs depending on the Bi‐concentration. Correspondingly, the free‐hole mobility (µh) decreases with increasing Bi‐concentration and eventually tends to increase for x > ∼8%. The incorporation of Bi in GaAs also induces the formation of acceptor levels, which we reveal by FIR absorption spectroscopy and Hall effect measurements. The Bi‐induced acceptors are characterized by an exceedingly high value of the effective ground‐state g‐factor (geff ∼ 15) and are responsible for the increasing p‐type conductivity observed in nominally undoped Ga(AsBi) alloys with increasing Bi‐concentration.