We report a strong shift of the plasma resonance in highly-doped GaAs/InGaAs core/shell nanowires (NWs) for intense infrared excitation observed by scattering-type scanning near-field infrared microscopy. The studied NWs show a sharp plasma resonance at a photon energy of about 125 meV in the case of continuous wave excitation by a CO2 laser. Probing the same NWs with the pulsed free-electron laser with peak electric field strengths up to several 10 kV cm-1 reveals a power-dependent redshift to about 95 meV and broadening of the plasmonic resonance. We assign this effect to a substantial heating of the electrons in the conduction band and subsequent increase of the effective mass in the nonparabolic Γ-valley.