Many astrophysical phenomena, including pulsars and short gamma-ray bursts, are associated with the extremely strong magnetic fields present in neutron stars and neutron star mergers. While the ideal magnetohydrodynamic approximation, which assumes infinite conductivity, provides an excellent description of the neutron-star interior, it cannot capture non-ideal processes such as Ohmic dissipation and magnetic reconnection. To overcome this limitation, we present an extension of the numerical-relativity code BAM incorporating a resistive general-relativistic magnetohydrodynamic (GRMHD) description. We validate the new implementation through an extensive suite of special-relativistic magnetohydrodynamic benchmark tests and by performing stable simulations of isolated and binary neutron star systems. For the latter, we investigate the impact of finite conductivity on magnetic-field amplification, mass ejection, and non-ideal GRMHD effects. In particular, we find that the component of the electric field parallel to the magnetic field, which is zero in the ideal case, can reach up to 10% of the total electric field strength. This highlights the potential importance of non-ideal effects for accurately modeling the long-term evolution of post-merger remnants, particularly in low-density regions. Although the present study is restricted to simplified piecewise-polytropic equations of state, it demonstrates the capabilities of the new resistive GRMHD framework and paves the way for future investigations employing more realistic microphysics.