In this paper, we study the one-loop induced photon’s effective action in the very special relativity electrodynamics in (2 + 1) spacetime (VSR –QED 3). Due to the presence of new nonlocal couplings resulting from the VSR gauge symmetry, we have additional graphs contributing to the ⟨ AA⟩ and ⟨ AAA⟩ amplitudes. From these contributions, we discuss the VSR generalization of the Abelian Maxwell–Chern–Simons Lagrangian, consisting in the dynamical part and the Chern–Simons-like self-couplings, respectively. We use the VSR–Chern–Simons electrodynamics to discuss some non-Ohmic behavior on topological materials, in particular VSR effects on Hall’s conductivity. In the dynamical part of the effective action, we observe the presence of a UV/IR mixing, due to the entanglement of the VSR nonlocal effects to the quantum higher-derivative terms. Furthermore, in the self-coupling aspect, we verify the validity of the Furry’s theorem in the VSR –QED 3 explicitly.