The association of Cepheid pulsation with traveling shocks and therefore X-ray non-thermal emission has been postulated for decades. Here we report the possibly first evidence of shock-accelerated GeV electrons emitted at the archetype classical Cepheid star $\delta$-Cep observed by XMM-Newton and Chandra. We jointly analyse the XMM-Newton thermal and non-thermal components of the time-resolved X-ray spectra prior to, during and after an X-ray enhancement. A comparison of the time scales of the diffusive particle acceleration at shocks with energy loss time scales 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. A 100-fold amplification of the magnetic field via turbulent dynamo at the shock propagating through density inhomogeneities in the stellar corona would be required for the synchrotron emission to dominate over the IC; however, such amplification is unlikely for the expected low Mach number shock produced. The lack of time-correlation between radio synchrotron and stellar pulsation contributes to make synchrotron emission as an unlikely mechanism for the flux enhancement. Current observations cannot rule out a high-flux two-temperature thermal spectrum with a negligible non-thermal component.