The cross section of the $^{65}\mathrm{Cu}(n,p)^{65}\mathrm{Ni}$ reaction was studied experimentally at three different neutron energies using an activation technique. The quasimonoenergetic neutrons were produced via the $^{7}\mathrm{Li}(p,n)$ reaction at the 14UD BARC-TIFR Pelletron facility in Mumbai, India. Al monitor foils along with Cu samples were activated to determine the incident neutron flux. The activities of the reaction products were measured using a high resolution high purity germanium spectrometry system. Statistical model calculations were performed using the reaction codes talys (ver. 1.9) and empire (ver. 3.2.3) from the reaction threshold to the neutron energy of 25 MeV. Additionally, the effects of various combinations of the theoretical nuclear level densities (NLDs), optical model potentials (OMPs), preequilibrium models (PEs), and \ensuremath{\gamma}-ray strength functions (\ensuremath{\gamma} SFs) were considered for the reproduction of experimental data. The input parameters needed in theoretical calculations to reproduce the present and previous measurements were taken from the RIPL-3 database. The present results are compared with the previous measurements, with the latest evaluations of the ENDF/B-VIII.0, JEFF-3.3, JENDL-4.0/HE, CENDL-3.2, TENDL-2019, and FENDL-3.2 libraries, and with the theoretically calculated values based on talys and empire codes. Furthermore, the cross section of the $^{65}\mathrm{Cu}(n,p)^{65}\mathrm{Ni}$ reaction was estimated within the neutron energies of 14--15 MeV using different systematic formulas. These estimated cross sections by various systematic formulas were compared with the available experimental data. The present data will help to understand the nuclear reaction theory (models) in higher energy regions and improve the evaluated nuclear data evaluation that is needed for fundamental nuclear applications.