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A manifestly N=2 supersymmetric completion of the four-dimensional NambuGoto-Born-Infeld action, which is self-dual with respect to electric-magnetic duality, is constructed in terms of an abelian N=2 superfield strength W in the conventional N=2 superspace. A relation to the known N=1 supersymmetric Born-Infeld action in N=1 superspace is established. The action found can be considered either as the Goldstone action associated with a partial breaking of N=4 supersymmetry down to N=2, with the N=2 vector superfield being a Goldstone field, or, equivalently, as the gauge-fixed superfield action of a D-3-brane in flat six-dimensional ambient spacetime.

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