$({\alpha }^{\mathrm{\prime }}{)}^4$ corrections to the N=2 supersymmetric Born-Infeld action

We consider the N=2 supersymmetric Born-Infeld action and compute one-loop divergences quantizing the theory in N=1 superspace. We find that in the presence of non constant curvature the theory is not renormalizable. The structure of the $({\alpha }^{\mathrm{\prime }}{)}^4$ counterterm, proportional to derivatives of the curvature, is consistent with effective action calculations from superstring theory.