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Within linear-response theory we derive a response function that thoroughly accounts for the in?uence of elastic scattering and is valid beyond the long-wavelength limit. We use the theory to evaluate the polarization function and the conductivity in metallic armchair graphene nanoribbons (AGNRs) in the Lindhard approximation for intra-band and inter-band transitions and for a relaxation-time t that is not constant. We obtain a logarithmic behaviour in the scattering-independent polarization function not only for intra-band transitions, as is usually the case for one-dimensional systems, but also for inter-band transitions. Modifying the screening wave vector and the impurity density in the long-wavelength limit strongly in?uences the relaxation time. In contrast, for large wave vectors, this modi?cation leads to a conservative value of t . We show that the imaginary part of the impurity-dependent conductivity varies with the wave vector while its scattering-independent part exists only for a single value of the wave vector.