  Radiofrequency (RF) electron guns operating at high accelerating gradients
offer a pathway to producing bright electron bunches. Such beams are expected
to revolutionize many areas of science: they could form the backbone of
next-generation compact x-ray free-electron lasers or provide coherent
ultrafast quantum electron probes. We report on the experimental demonstration
of an RF photoemission electron source supporting an accelerating field close
to 400~MV/m at the photocathode surface. The gun was operated in an RF
transient mode driven by short $\sim 9$~ns X-band (\SI{11.7}{\giga\hertz}) RF
pulses. We did not observe any major RF breakdown or significant dark current
over a three-week experimental run at high accelerating fields. The
demonstrated paradigm provides a viable path to forming relativistic electron
beams with unprecedented brightness.
