A theory of global geodesic acoustic modes with short wavelengths based on the framework of an electrostatic gyro-fluid has been developed, in which the more thorough sorts of finite-ion-Larmor-radius effects are encompassed than in the two-fluid model (Wang et al 2021 Nucl. Fusion61 106024), while the radial differential equation of the eigenmodes still comes up in a compact and analytically tractable form. By solving the corresponding boundary value problem numerically with typical equilibrium profiles, a multitude of global geodesic acoustic modes with quantized eigen-frequencies have been found. In particular, as the consequence of the plasma model improvement, we have obtained multiple edge-localized global geodesic acoustic modes with the features resembling those observed experimentally.