Bi-containing III-V semiconductors constitute an exciting class of metastable compounds with wide-ranging potential optoelectronic and electronic applications. However, the growth of III-V-Bi alloys requires group-III-rich growth conditions, which pose severe challenges for planar growth. In this work, we exploit the naturally-Ga-rich environment present inside the metallic droplet of a self-catalyzed GaAs nanowire to synthesize metastable GaAs/GaAs$_{1-\text{x}}$Bi$_{\text{x}}$ axial nanowire heterostructures with high Bi contents. The axial GaAs$_{1-\text{x}}$Bi$_{\text{x}}$ segments are realized with molecular beam epitaxy by first enriching only the vapor-liquid-solid (VLS) Ga droplets with Bi, followed by exposing the resulting Ga-Bi droplets to As$_2$ at temperatures ranging from 270 to 380$\,^{\circ}$C to precipitate GaAs$_{1-\text{x}}$Bi$_{\text{x}}$ only under the nanowire droplets. Microstructural and elemental characterization reveals the presence of single crystal zincblende GaAs$_{1-\text{x}}$Bi$_{\text{x}}$ axial nanowire segments with Bi contents up to (10$\pm$2)$\%$. This work illustrates how the unique local growth environment present during the VLS nanowire growth can be exploited to synthesize heterostructures with metastable compounds.