A nonreciprocal magnon laser is proposed in a compound cavity optomagnonical system consisting of an yttrium iron garnet sphere coupled to a spinning resonator. On the basis of the magnon-induced Brillouin scattering process making it possible to achieve a magnon lasing action, the Fizeau light-dragging effect caused by the spinning of the resonator further results in significant modifications in the magnon gain and the threshold power of magnon lasing for different driving directions, and then a nonreciprocal magnon laser is realized. Especially, this nonreciprocal magnon laser is highly tunable by the spinning speed and the driving direction. Our work provides an experimentally feasible pathway for manipulating spin-wave excitations and may find intriguing phenomena at the crossroad between spintronics of the magnet and nonreciprocal optics.