This paper delves into the workspace trajectory tracking control problem of space manipulators working in free-floating mode encountering external disturbance and actuator saturation. For the sake of mitigating tracking error overshoot, the asymmetric prescribed performance function is employed to keep tracking errors within user-defined performance boundaries. In order to conduct an output feedback controller, an extended state observer is developed for estimating the unmeasurable velocity and counteracting external disturbance. Then, an auxiliary control signal is introduced to deal with the actuator saturation. The Lyapunov synthesis ensures the uniform boundedness of all closed-loop signals. Finally, comparative numerical simulations are conducted to validate the efficacy of the developed controller.