Scanning tunneling microscopy and spectroscopy have been employed to investigate the superconductivity in single unit-cell FeSe nanoflakes on SrTiO3 substrates. We find that the differential conductance dI/dV spectra are spatially nonuniform and fluctuate within the flakes as their area is reduced to below ∼150 nm2. An enhancement in the superconductivity-related gap size as large as 25% is observed. The superconductivity behavior disappears when the FeSe nanoflakes reduce to ∼40 nm2. Compared to a previous report [Wang et al., Chin. Phys. Lett. 29, 037402 (2012)], the gap is asymmetric relative to the Fermi energy EF. All the features, particularly the fluctuating gap and quenched superconductivity, could be accounted for by quantum size effects. Our study helps to understand nanoscale superconductivity in low-dimensional systems.