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We investigate the nonlinear optical response of a commercial extended-wavelength In$_{0.81}$Ga$_{0.19}$As photodetector. Degenerate two-photon absorption in the mid-infrared range is observed at room temperature using a quantum cascade laser emitting at $\lambda=4.5~\mu$m as the excitation source. From the measured two-photon photocurrent signal we extract a two-photon absorption coefficient $\beta^{(2)} = 0.6 \pm 0.2$ cm/MW, in agreement with the theoretical value obtained from the $E_g^{-3}$ scaling law. Considering the wide spectral range covered by extended-wavelength In$_x$Ga$_{1-x}$As alloys, this result holds promise for new applications based on two-photon absorption for this family of materials at wavelengths between 1.8 and 5.6 $\mu$m.