We introduce a new first-order formulation of worldvolume actions for p-branes with κ-supersymmetry. In this language, which involves more auxiliary fields compensated by more local symmetries, the action is provided by a very compact, simple and elegant formula applicable to any supergravity background. The κ-supersymmetry variation against which it is invariant is obtained from the bulk supersymmetries by means of a projector that has a simple expression in terms of the auxiliary fields. The distinctive feature of our formalism is that all fermion fields are hidden into the definition of the curvatures and the action is formally the same, in terms of these differential forms as it would be in a purely bosonic theory. Typically, our new formulation enables one to discuss the correct boundary actions for non-trivial supergravity Dp-brane bulk solutions such as the D3-brane solution on smooth ALE manifolds with flux recently constructed in the literature.