  We present the first complete calculation of the soft photon self-energy to
the next-to-leading order in a hot and/or dense ultrarelativistic plasma in
Quantum Electrodynamics (QED). The calculation is performed within the
real-time formalism utilizing dimensional regularization in $4-2\epsilon$
dimensions, while the result is reported including explicit $O(\epsilon)$ terms
in the zero-temperature limit. This information is required to extend the
perturbative calculation of the pressure of cold and dense QED matter to
partial next-to-next-to-next-to-leading order in a weak-coupling expansion,
reported in a companion paper. These results pave the way for a similar future
calculation in Quantum Chromodynamics.
