This article proposes a novel adaptive integral terminal third-order finite-time sliding-mode control strategy dedicated to motion tracking control of a piezoelectric-driven nanopositioning system. One uniqueness of the reported controller lies in that it generates finite-time convergence along with an inherent chattering-free control action. The model uncertainty involving the hysteresis nonlinearity and unmodeled dynamics is estimated through the perturbation estimation technique. Adaptive rules are established to update the control gains without acquiring the upper boundary of the estimation error. The stability of the presented third-order adaptive integral terminal sliding-mode control strategy has been proved based upon the Lyapunov analysis. It enables the achievement of third-order sliding-mode precision and prespecification of the tracking error boundary. The controller performance is verified by performing both simulation and experimental investigations. Results reveal the fine robust tracking performance of the developed controller for the nanopositioning system in the presence of model uncertainty and external disturbance.