We present BeppoSAX observations of the southern S1 region in the European Large-Area Infrared Space Observatory (ISO) Survey (ELAIS). These observations cover an area of similar to1.7 deg(2) and reach an on-axis (similar to0.7 deg(2)) 2-10 keV (hard X-ray, or HX) sensitivity of similar to 10(-13) ergs s(-1) cm(-2). This is the first HX analysis of an ISOCAM survey. We detect nine sources with a signal-to-noise ratio SNRHX > 3, four additional sources with a 1.3-10 keV (total X-ray, or T) SNRT > 3, and two additional sources that seem to be associated with QSOs having SNRT > 2.9. The number densities of the SNRHX > 3 sources are consistent with the ASCA and BeppoSAX log N-log S functions. Six BeppoSAX sources have reliable ISOCAM 15 mum counterparts within similar to 60". All these ISOCAM sources have optical counterparts of R < 20 mag. Five of these sources have been previously optically classified, four as QSOs and one as a broad absorption line (BAL) QSO at z = 2.2. The remaining unclassified source has X-ray and photometric properties consistent with those of a nearby Seyfert galaxy. One further HX source has a 2.6 <sigma> ISOCAM counterpart associated with a galaxy at z = 0.325. If this ISOCAM source is real, the HX/MIR properties suggest either an unusual QSO or a cD cluster galaxy. We have constructed MIR and HX spectral energy distributions to compute the expected HX/MIR ratios for these classes of objects up to z = 3.2 and assess the HX/MIR survey depth. The BAL QSO has an observed X-ray softness ratio and HX/MIR flux ratio similar to those of QSOs but different from those found for low-redshift BAL QSOs. This difference can be explained in terms of absorption, and it suggests that high-redshift BAL QSOs should be comparatively easy to detect in the HX band, allowing their true fraction in the high-redshift QSO population to be determined. The QSOs cover a wide redshift range (0.4 <z <2.6) and have HX/MIR flux ratios consistent with those found for nearby IRAS and optically selected Palomar-Green QSOs. This suggests that MIR-selected QSOs of R<20 mag come from the same population as optically selected QSOs. We confirm this with a comparison of the B/MIR flux ratios of MIR and blue-band-selected QSOs.
