We propose and experimentally demonstrate a fast Berry-phase gate, which is implemented by picosecondtimescale optical pulses to make the qubit system of atomic clock states adiabatically evolve on a closed loop. The characteristic features of the proposed gate are gate speed and robustness against control fluctuations, which can potentially resolve the decoherence and reliability issues in quantum information processing, at the same time. The experiment is conducted with two linearly polarized, chirped optical pulses, interacting with five single rubidium atoms simultaneously in an array of optical tweezer dipole traps, to demonstrate the proposed picosecond-timescale clock-state gates. The robustness of the qubit rotation angle δ /δA = 1.5% is achieved with respect to the laser intensity (of pulse area A) fluctuation.