  The collision of ultrashort high-density $e^-$ or $e^-$ and $e^+$ beams at
10s of GeV, to be available at the FACET II and in laser wakefield accelerator
experiments, can produce highly collimated $\gamma$ rays (few GeVs) with peak
brilliance of $10^{27}~\mathrm{ph/s~mm^2 mrad^2 0.1\% BW}$ and up to $10^5$
nonlinear Breit-Wheeler pairs. We provide analytical estimates of the photon
source properties and of the yield of secondary pairs, finding excellent
agreement with full-scale 3D self-consistent particle-in-cell simulations that
include quantum electrodynamics effects. Our results show that beam-beam
collisions can be exploited as secondary sources of $\gamma$ rays and provide
an alternative to beam-laser setups to probe quantum electrodynamics effects at
the Schwinger limit.
