Copper-64 is a radioisotope of medical interest that could be used for positron emission tomography imaging and targeted radiotherapy of cancer. In this work, we investigated the possibility of producing the ~(64)Cu isotope through a ~(65)Cu(γ,n) reaction using high-intensity c-beams produced at the Extreme Light Infrastructure- Nuclear Physics facility (ELI-NP). The specific activity for ~(64)Cu was obtained as a function of target geometry, irradiation time, and electron beam energy, which translates into γ-beam energy. Optimized conditions for the generation of ~(64)Cu isotopes at the ELI-NP were discussed. We estimated that an achievable saturation specific activity is of the order of 1-2 mCi/g for thin targets (radius 1-2 mm, thickness 1 cm) and for a γ-beam flux of 10~(11) s~(-1). Based on these results, the ELI-NP could provide great potential for the production of some innovative radioisotopes of medical interest in sufficient quantities suitable for nuclear medicine research.