Context. Compact star-forming galaxies were dominant galaxy types in the early Universe. Blueberry galaxies (BBs) represent their local analogues, being very compact and having intense star formation. Aims. Motivated by high X-ray emission recently found in other analogical dwarf galaxies, called Green Peas, we probed the X-ray properties of BBs to determine if their X-ray emission is consistent with the empirical laws for star-forming galaxies. Methods. We performed the first X-ray observations of a small sample of BBs with the XMM-Newton satellite. Spectral analysis for detected sources and upper limits measured via Bayesian-based analysis for very low-count measurements were used to determine the X-ray properties of our galaxy sample. Results. Clear detection was obtained for only two sources, with one source exhibiting an enhanced X-ray luminosity to the scaling relations. For the remaining five sources, only an upper limit was constrained, suggesting BBs to be rather underluminous as a whole. Our analysis shows that the large scatter cannot be easily explained by the stochasticity effects. While the bright source is above (and inconsistent with) the expected distribution at almost the 99% confidence level, the upper limits of the two sources are below the expected distribution. Conclusions. These results indicate that the empirical relations between the star formation rate, metallicity, and X-ray luminosity might not hold for BBs with uniquely high specific star formation rates. One possible explanation could be that the BBs may not be old enough to have a significant X-ray binary population. The high luminosity of the only bright source can then be caused by an additional X-ray source, such as a hidden active galactic nucleus or more extreme ultraluminous X-ray sources.