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We look for a classical double copy structure between gravity andelectrodynamics by connecting the descriptions of the scattering of two pointmasses, and of two point charges, in terms of perturbative (post-Minkowskian orpost-Lorentzian) expansions. We do so by recasting available analyticalinformation within the effective-one-body formalism using Kerr-Schild gauges inboth cases. Working at the third perturbative level, we find that the usuallinear relation (holding in the probe limit) between the dimensionless electricpotential, $\tilde{\phi}= \frac{G M}{e_1 e_2} \phi^{\rm el}$, and theSchwarzschildlike gravitational one, $\Phi^{\rm grav}$, is deformed, in thecomparable-mass, comparable-charge, case, into a nonlinear relation whichbecomes universal in the high energy limit: $\Phi^{\rm grav}=2\tilde{\phi}-5\tilde{\phi}^2 + 18\tilde{\phi}^3$.

Double copy, Kerr-Schild gauges and the Effective-One-Body formalism