A reconfigurable intelligence surface (RIS) consists of a large-scale array of optoelectronic devices that can modulate the electromagnetic characteristics, such as the amplitude and phase, of an incident electromagnetic wave to obtain a customized reflected beam and improve the wireless positioning service problem in non-line-of-sight (NLOS) environments. In this study, a non-orthogonal multiple access (NOMA) positioning system with intelligent reflective surfaces is constructed, which uses multiple RISs deployed in the air to reflect the signals transmitted from the base station to the user for easy reception. In addition, the RISs use pseudo-random sequences to remodulate the signals transmitted from the base station and then reflect them back to the user, making it easier to identify the signals reflected from different RISs. Finally, the user estimates multiple signal angles of arrival (AOA) using the multiple signal classification (MUSIC) algorithm to obtain the positioning coordinates. The simulation results show that the method proposed in this study can achieve centimeter-level positioning accuracy in NLOS environments.