We present a nonextensive version of the QCD-based Nambu–Jona-Lasinio (NJL) model of a many-body field theory describing the behavior of strongly interacting matter. It is based on the nonextensive generalization of Boltzmann–Gibbs (BG) statistical mechanics used in the NJL model, which was taken in the form proposed by Tsallis characterized by a dimensionless nonextensivity parameter q (for q → 1 one recovers the usual BG case). This new phenomenological parameter accounts summarily for all possible effects resulting in a departure from the conditions required by the application of the BG approach, and allows for a simple phenomenological check of the sensitivity of the usual NJL model to such effects (in particular to fluctuations of temperature and correlations in a system of quarks). As an example, we discuss the sensitivity of such a q-NJL model to the departures from the NJL form, both for q > 1 and q < 1 cases, for such observables as the temperature dependences of chiral symmetry restoration, masses of π and σ mesons and characteristic features of spinodal decomposition.