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We report the first detection of hard (>10 keV) X-ray emission simultaneous with gamma-rays in a nova eruption. Observations of the nova V5855 Sgr carried out with the NuSTAR satellite on Day 12 of the eruption revealed faint, highly absorbed thermal X-rays. The extreme equivalent hydrogen column density toward the X-ray emitting region (˜3 × 10^24 cm^-2) indicates that the shock producing the X-rays was deeply embedded within the nova ejecta. The slope of the X-ray spectrum favors a thermal origin for the bulk of the emission, and the constraints of the temperature in the shocked region suggest a shock velocity compatible with the ejecta velocities inferred from optical spectroscopy. While we do not claim the detection of nonthermal X-rays, the data do not allow us to rule out an additional, fainter component dominating at energies above 20 keV, for which we obtained upper limits. The inferred luminosity of the thermal X-rays is too low to be consistent with the gamma-ray luminosities if both are powered by the same shock under standard assumptions regarding the efficiency of nonthermal particle acceleration and the temperature distribution of the shocked gas.