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Rapid X-ray phase-dependent flux enhancement in the archetype classical Cepheid starδCep was observed by XMM-Newton and Chandra. We jointly analyze thermal and nonthermal components of the time-resolved X-ray spectra prior to, during, and after the enhancement. A comparison of the timescales 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 ∼GeV energies, and with Inverse Compton (IC) emission from the UV stellar background leading to the observed X-ray enhancement. The index of the nonthermal 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 ∼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 nonthermal component, this event might confirm for the first time the association of Cepheids pulsation with shock-accelerated GeV electrons.