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For high-performance trajectory tracking at the nanometer scales, this paper presents a new fast terminal sliding mode controller, which combines a recursive integer-order non-singular high-order sliding manifold and a fractional-order fast fixed-time reaching law to ensure globally fast convergence, and adopts a time-delay-estimation (TDE) based disturbance estimator deeming the designed controller robust to parameter uncertainty. Stability of the designed controller is verified through the Lyapunov framework, where the full analyses of convergence region and settling time are also presented. The tracking performance is experimentally verified on a piezo-stack driven nano-positioning platform. To showcase the performance improvements, measured closed-loop performance of the proposed controller is contrasted with those obtained using three benchmark control approaches namely the basic Proportional-Integral-Derivative (PID), the popular Positive Position Feedback with Integral action (PPF+I), and the traditional linear sliding mode controller (LSMC).