Gaia Data Release 3. The first Gaia catalogue of eclipsing-binary candidates

We present the first Gaia catalogue of eclipsing binary candidates released in Gaia DR3, describe its content, provide tips for its usage, estimate its quality, and show illustrative samples. The catalogue contains 2,184,477 sources with G magnitudes up to 20 mag. Candidate selection is based on the results of variable object classification performed within the Gaia Data Processing and Analysis Consortium, further filtered using eclipsing binary-tailored criteria based on the G light curves. To find the orbital period, a large ensemble of trial periods is first acquired using three distinct period search methods applied to the cleaned G light curve. The G light curve is then modelled with up-to two Gaussians and a cosine for each trial period. The best combination of orbital period and geometric model is finally selected using Bayesian model comparison based on the BIC. A global ranking metric is provided to rank the quality of the chosen model between sources. The catalogue is restricted to orbital periods larger than 0.2 days. About 530,000 of the candidates are classified as eclipsing binaries in the literature as well, out of ~600,000 available crossmatches, and 93% of them have published periods compatible with the Gaia periods. Catalogue completeness is estimated to be between 25% and 50%, depending on the sky region, relative to the OGLE4 catalogues of eclipsing binaries towards the Galactic Bulge and the Magellanic Clouds. The analysis of an illustrative sample of ~400,000 candidates with significant parallaxes shows properties in the observational HR diagram as expected for eclipsing binaries. The subsequent analysis of a sub-sample of detached bright candidates provides further hints for the exploitation of the catalogue. The orbital periods, light curve model parameters, and global rankings are all published in the catalogue with their related uncertainties where applicable.