The free-float space manipulator (FFSM) control problem subject to coupling dynamics, uncertainties, and input saturation constraints is investigated in this paper. The screw-based dynamical model is established through the Lagrangian equation and Hamilton’s variational principle such that the posture of the rigid body can be described uniquely and globally. The derived dynamics is then reduced to a decoupled one in the light of the conservation of the momentum. On this basis, a novel anti-windup finite-time sliding-mode control strategy is investigated to guarantee the tracking performance in the joint space of FFSM, where a learning-based adaptive neural network is designed to estimate the uncertainties. The approximation error is inhibited by a tanh-type slow-varying adaptive law which is integrated into an auxiliary system to compensate for input saturation nonlinearities. With the contributions above, the stability of the proposed nonsingular terminal sliding-mode (NTSM) controller is comprehensively proved. Comparative simulations based on a 6-DOF FFSM confirm the validity of the proposed control scheme.