  The Kerr-Schild (KS) formalism is a powerful tool for constructing exact
solutions in general relativity. In this paper, we present a generalization of
the conventional KS formalism to double field theory (DFT) and supergravities.
We introduce a generalized KS ansatz for the generalized metric in terms of a
pair of null vectors. Applying this ansatz to the equations of motion of DFT,
we construct the generalized KS field equation. While the generalized KS
equations are quadratic in the fields, we show that it is possible to find
solutions by considering linear equations only. Furthermore, we construct a
Killing spinor equation under the generalized KS ansatz. Based on this
formalism, we show that the classical double copy structure, which represents
solutions of the Einstein equation in terms of solutions of the Maxwell
equation, can be extended to the entire massless string NS-NS sector. We
propose a supersymmetric classical double copy which shows that solutions of
the Killing spinor equation can be realized in terms of solutions of the BPS
equation of the supersymmetric Maxwell theory.
