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• The finite-time velocity-free problem of 6-DOF spacecraft control is considered. • A novel finite-time velocities observer is presented in the dual-quaternion form. • A new observer-based non-singular terminal sliding mode control law is proposed. • The finite-time stability of observer-alone and the system is strictly proved. • The control law can be applied to various tasks by designing the reference frame. This paper addresses the problem of spacecraft's finite-time six-degree-of-freedom(6-DOF) tracking control without velocity measurements. To estimate the unavailable incorporating angular and translational velocities (called the dual angular velocity), a novel observer with a simple structure and finite-time stability is presented in the dual quaternion frame. Then, combining with the estimated dual angular velocity, a continuous velocity-free 6-DOF control law based on the terminal sliding mode theory is presented to achieve the relative pose tracking objective within finite time, where the specially designed parameter ensures its non-singularity. In addition, by designing the desired reference frame, the proposed control law can be employed to various missions. With consideration of the observation errors, a rigorous proof is provided to demonstrate the finite-time stability of the observer-alone and the observer-based control law. Two typical numerical simulations, hovering and circumnavigation, are carried out to verify the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]