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Axions are hypothetical particles proposed to solve the strong CP problem in QCD and may constitute a significant fraction of the dark matter in the Universe. Axions are expected to be produced in neutron stars and subsequently decay, producing gamma-rays detectable by the Fermi Large Area Telescope (Fermi-LAT). Considering that light QCD axions, as opposed to axions $>1$eV, may travel a long range before they decay into gamma rays, neutron stars may appear as a spatially-extended source of gamma rays. We extend our previous search for gamma rays from axions, based on a point source model, to consider the neutron star as an extended source of gamma rays. The extended consideration of neutron stars' leads to higher sensitivity to searches for axions, as it will be shown. We investigate the spatial emission of gamma rays using phenomenological models of neutron star axion emission. We present models including the fundamental astrophysics and relativistic, extended gamma-ray emission from axions around neutron stars. A Monte Carlo simulation of the LAT gives us an expectation for the extended angular profile and spectrum. For a source of $\simeq$ 100 pc, we predict a mean angular spread of $\simeq 2^\circ$ with gamma-ray energies in the range 10-200 MeV. We demonstrate the feasibility of setting more stringent limits for axions in this mass range, excluding a range not probed by observations before. We consider projected sensitivities for mass limits on axions from J0108-1431. Based on the extended angular profile of the source, the expected sensitivity of the 95\% CL upper limit on the axion mass from J0108-1431 is $\lesssim$10 meV. The limit based on 7.9 years of Fermi-LAT data is 0.76 meV for an inner temperature of the neutron star of 20 MeV.