Random lasers are a class of devices in which feedback arises from multiple elastic scattering in a highly disordered structure, providing an almost ideal light source for artefact-free imaging due to achievable low spatial coherence. However, for many applications ranging from sensing and spectroscopy to speckle-free imaging, it is essential to have high-radiance sources operating in continuous-wave (CW). In this paper, we demonstrate CW operation of a random laser using an electrically pumped quantum-cascade laser gain medium in which a bi-dimensional (2D) random distribution of air holes is patterned into the top metal waveguide. We obtain a highly collimated vertical emission at ~3 THz, with a 430 GHz bandwidth, device operation up to 110 K, peak (pulsed) power of 21 mW, and CW emission of 1.7 mW. Furthermore, we show that an external cavity formed with a movable mirror can be used to tune a random laser, obtaining continuous frequency tuning over 11 GHz.