  Quantum non-Gaussianity, a more potent and highly useful form of
nonclassicality, excludes all convex mixtures of Gaussian states and Gaussian
parametric processes generating them. Here, for the first time, we conclusively
test quantum non-Gaussian coincidences of entangled photon pairs with the
CHSH-Bell factor $S=2.328\pm0.004$ from a single quantum dot with a depth up to
$0.94\pm 0.02$ dB. Such deterministically generated photon pairs fundamentally
overcome parametric processes by reducing crucial multiphoton errors. For the
quantum non-Gaussian depth of the unheralded (heralded) single-photon state, we
achieve the value of $8.08\pm0.05$ dB ($19.06\pm0.29$ dB). Our work
experimentally certifies the exclusive quantum non-Gaussianity properties
highly relevant for optical sensing, communication and computation.
