$N=4$ supersymmetric $d=3$ nonlinear electrodynamics

We construct a new off-shell $N=4$, $d=3$ nonlinear vector supermultiplet. The irreducibility constraints for the superfields leave in this supermultiplet the same component content as in the ordinary linear vector supermultiplet. We present the most general sigma-model type action for the $N=4$, $d=3$ electrodynamics with the nonlinear vector supermultiplet, which despite the nonlinearity of the supermultiplet may be written as an integral over a chiral superspace. This action shares the most important properties with its linear counterpart. We also perform the dualization of the vector component into a scalar one and find the corresponding $N=4$, $d=3$ supersymmetric action which describes new hyper-Kähler sigma-model in the bosonic sector.