A strong negative photoconductivity was identified in thin film networks of single-walled carbon nanotubes using optical pump, THz probe spectroscopy. The films were controllably doped, using either adsorption doping with different p-type dopant concentrations or ambipolar doping using an ionic gate. While doping enhanced the THz conductivity and increased the momentum scattering rate, interband photoexcitation lowered the spectral weight and reduced the momentum scattering rate. This negative THz photoconductivity was observed for all doping levels, regardless of the chemical potential, and decayed within a few picoseconds. The strong many-body interactions inherent to these 1D conductors led to trion formation under photoexcitation, lowering the overall conductivity of the carbon nanotube network. The large amplitude of negative THz photoconductivity and the tunability of its recovery time with doping offer promise for spectrally wide-band ultrafast devices, including THz detectors, polarizers, and modulators.