This article considers millimeter-wave (mmWave) communications in a full-duplex (FD) unmanned aerial vehicle (UAV) system with multiple non-orthogonal multiple access (NOMA) users. In particular, a UAV operates in an FD transmission mode and acts as a decode-and-forward (DF) relay to aid communications between a ground base station and multiple ground users. Unlike previous works on the UAV-aided NOMA systems, where the specific carrier frequencies are often ignored, this work investigates the effect of a specific carrier frequency in the mmWave band and the channel model proposed for the fifth and beyond generations (5G and B5G) of mobile communications. The closed-form expressions of outage probability (OP) and achievable data rate (ADR) of the FD-UAV-NOMA system with mmWave communications (called as the MW-FD-UAV-NOMA system) are successfully derived over Nakagami-m fading channels. Numerical results clearly indicate the strong effects of the residual self-interference (RSI), high frequency, far distances between transmitters and receivers, wide bandwidth, and the altitude of the UAV on the OP and ADR of the MW-FD-UAV-NOMA system. Since the mmWave communications are used, an increase in the frequency, distances, bandwidth, and/or altitude of the UAV will significantly reduce the performance of the MW-FD-UAV-NOMA system. Importantly, when the UAV flies nearer the base station, the performance of the MW-FD-UAV-NOMA system can be better than when the UAV flies nearer the users. Finally, the exactness of the obtained expressions is confirmed via Monte-Carlo simulations.