Field-effect phototransistors were fabricated based on individual carbon nanotubes (CNTs) sensitized by CH3NH3PbI3 nanowires (MAPbI(3)NWs). These devices represent light responsivities of R = 7.7 x 10(5) A W-1 under low-lighting conditions in the nW mm(-2) range, unprecedented among CNT-based photodetectors. At high incident power (similar to 1 mW mm(-2)), light soaking results in a negative photocurrent, turning the device insulating. We interpret the phenomenon as a result of efficient free photoexcited charge generation and charge transfer of photoexcited holes from the perovskite to the carbon nanotube. The charge transfer improves conductance by increasing the number of carriers, but leaves electrons behind. At high illumination intensity their random electrostatic potential quenches mobility in the nanotube.