This study proposes a new asynchronous Multi-Rate Multiple-Access Differential Chaos Shift Keying (MRMA-DCSK) scheme, ensuring significant data rates for all users. This scheme assigns a unique chaos sequence with a different length to each user. During the first data transmission period, each user transmits the chaos sequence as the reference signal, followed by multiple data bits by sharing the same reference signal in subsequent periods. The proposed scheme affects the bit-error-rate (BER) performance with the number of users, data rate related parameters (L), and length of chaos signals. The simulation results are verified by the derived analysis and show that the proposed scheme achieves higher data rates (from 1/2 to L/L+1) than a conventional scheme while enhancing bit-error-rate (BER) performance.