A high precision calibration of the nonlinearity in the energy response of the Daya Bay Reactor Neutrino Experiments antineutrino detectors is presented in detail. The energy nonlinearity originates from the particle-dependent light yield of the scintillator and charge-dependent electronics response. The nonlinearity model is constrained by $gamma$ calibration points from deployed and naturally occurring radioactive sources, the $beta$ spectrum from $^{12}$B decays, and a direct measurement of the electronics nonlinearity with a new flash analog-to-digital converter readout system. Less than 0.5% uncertainty in the energy nonlinearity calibration is achieved for positrons of kinetic energies greater than 1 MeV.
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