jats:titleAjats:scbstract</jats:sc> </jats:title>jats:pWe give a prescription for jats:inline-formulajats:alternativesjats:tex-math$$ \mathcal{N} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> mml:miN</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> = 1 supersymmetrization of any (four-dimensional) nonlinear electrodynamics theory with a Lagrangian density satisfying a convexity condition that we relate to semi-classical unitarity. We apply it to the one-parameter ModMax extension of Maxwell electrodynamics that preserves both electromagnetic duality and conformal invariance, and its Born-Infeld-like generalization, proving that duality invariance is preserved. We also establish superconformal invariance of the superModMax theory by showing that its coupling to supergravity is super-Weyl invariant. The higher-derivative photino-field interactions that appear in any supersymmetric nonlinear electrodynamics theory are removed by an invertible nonlinear superfield redefinition.</jats:p>