  Rapid X-ray phase-dependent flux enhancement in the archetype classical
Cepheid star $\delta$~Cep was observed by XMM-Newton and Chandra. We jointly
analyse thermal and non-thermal components of the time-resolved X-ray spectra
prior to, during and after the enhancement. A comparison of the time scales of
shock particle acceleration and energy losses is consistent with the scenario
of a pulsation-driven shock wave traveling into the stellar corona and
accelerating electrons to $\sim$ GeV energies and with Inverse Compton (IC)
emission from the UV stellar background leading to the observed X-ray
enhancement. The index of the non-thermal IC photon spectrum, assumed to be a
simple power-law in the $[1-8]$ keV energy range, radially integrated within
the shell $[3 - 10]$ stellar radii, is consistent with an enhanced X-ray
spectrum powered by shock-accelerated electrons. An unlikely $\sim$100-fold
amplification { via turbulent dynamo} of the magnetic field at the shock
propagating through density inhomogeneities in the stellar corona is required
for the synchrotron emission to dominate over the IC; the lack of
time-correlation between radio synchrotron and stellar pulsation contributes to
make synchrotron as an unlikely emission mechanism for the flux enhancement.
Although current observations cannot rule out a high-flux two-temperature
thermal spectrum with a negligible non-thermal component, this event might
confirm for the first time the association of Cepheids pulsation with
shock-accelerated GeV electrons.
