A novel grant-free random access scheme is proposed to support massive connectivity with low access delay and overhead for machine-to-machine communication in massive multiple-input-multiple-output systems. This scheme allows all active user equipments (UEs) to transmit their pilots and uplink messages via the same time-frequency resource and performs the joint active UEs detection and uplink message decoding without channel estimation in one shot by utilizing the proposed ensemble independent component analysis (EICA) decoding algorithm. We call the proposed scheme the EICA-based pilot random access (EICA-PA). We analyze the successful access probability, probability of missed detection, and uplink throughput of the EICA-PA scheme. Numerical results show that the EICA-PA scheme significantly improves the successful access probability and uplink throughput, decreases missed detection probability and provides low-frame error rate at the same time.