A continuum manipulator is a new type of bionic manipulator. Due to its high flexibility and adaptability, it has shown significant application potential in many fields. This paper focuses on the tracking control problem for a continuum manipulator with two degrees of freedom (DOF). An anti-saturation fixed-time sliding mode control strategy is developed for the 2-DOF continuum manipulator in the presence of input saturation. Firstly, the dynamic model of the continuum manipulator is given. Secondly, an adaptive input saturation compensator auxiliary system is constructed to compensate for the impact of input saturation. And then, the tracking controllers are designed based on fixed-time stability theory. Our proposed control strategy can effectively achieve the fixed-time tracking control objective of the continuum manipulator when there are saturation limits on the control inputs. And the effectiveness of the strategy is verified through numerical simulation experiments.