This paper presents a fixed-time distributed cooperative control scheme based on non-singular fast terminal sliding mode to solve the problems of time-varying marine environment disturbances, model parameter uncertainties, unavailable velocity measurements and input saturation in the cooperative control process for multiple surface vessels. The scheme ensures the global fixed time stability of the cooperative control system. Firstly, a fixed-time extended state observer is designed to estimate the velocities and lumped disturbances, including time-varying marine environmental disturbances and model parameter uncertainties. Secondly, a novel fixed-time auxiliary dynamic system is designed to deal with input saturation. Thirdly, a distributed anti-saturation sliding mode controller is proposed based on the fixed-time non-singular fast terminal sliding mode manifold to eliminate the singularity of the system, and make the upper bound of convergence time independent of the initial state of the system. Finally, in order to avoid using the acceleration for the adjacent vessel in the formation control law, an improved uniform robust exact differentiator is designed to estimate the acceleration information. The simulation results show that the proposed scheme is effective.