We present a mid-infrared quantum cascade laser (QCL) with a sub-10 kHz full width at half-maximum linewidth (at 1 s integration time) achieved by stabilization to a free-space optical delay line. The linear range in the center of a fringe detected at the output of an imbalanced Mach-Zehnder interferometer implemented with a short free-space pathlength difference of only 1 m is used as a frequency discriminator to detect the frequency fluctuations of the QCL. Feedback is applied to the QCL current to lock the laser frequency to the delay line. The application of this method in the mid-infrared is reported for the first time, to the best of our knowledge. By implementing it in a simple self-homodyne configuration, we have been able to reduce the frequency noise power spectral density of the QCL by almost 40 dB below 10 kHz Fourier frequency, leading to a linewidth reduction by a factor of almost 60 compared to the free-running laser. The present limits of the setup are assessed and discussed.