  We construct an off-shell N=3 supersymmetric extension of the abelian D=4
Born-Infeld action starting from the action of supersymmetric Maxwell theory in
N=3 harmonic superspace. A crucial new feature of the N=3 super BI action is
that its interaction part contains only terms of the order 4k in the N=3
superfield strengths. The correct component bosonic BI action arises as the
result of elimination of auxiliary tensor field which is present in the
off-shell N=3 vector multiplet in parallel with the gauge field strength. In
this new Legendre-type representation, the bosonic BI action is fully specified
by a real function of the single variable quartic in the auxiliary tensor
field. The generic choice of this function amounts to a wide set of self-dual
nonlinear extensions of the Maxwell action. All of them admit an off-shell N=3
supersymmetrization.
