This article investigates the distributed containment control problem for high‐order stochastic nonstrict‐feedback nonlinear multi‐agent systems. To overcome the difficulty of unknown control input gain caused by the asymmetric hysteretic quantization and actuator fault, the properties of Nussbaum function are employed and an auxiliary virtual controller is designed, which does not need to estimate the bounds of the parameters of asymmetric hysteretic quantizer and efficiency factors. Moreover, the unknown terms are estimated by introducing the radial basis function neural networks. Based on the input‐driven filter, an adaptive containment control scheme is designed by using the Nussbaum function technique, the backstepping approach, and the dynamic surface control method. It is shown that the proposed control method can guarantee that all the signals are semi‐globally uniformly ultimately bounded, and the containment control performance can be achieved. Finally, some simulation results are presented to illustrate the effectiveness of the proposed approach.