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Context. The astrometric discovery of sub-stellar mass companions orbiting stars is exceedingly hard due to the required sub-milliarcsecond precision, limiting the application of this technique to only a few instruments on a target-per-target basis and to the global astrometry space missions H IPPARCOS and Gaia . The third Gaia data release ( Gaia DR3) includes the first Gaia astrometric orbital solutions whose sensitivity in terms of estimated companion mass extends down to the planetary-mass regime. Aims. We present the contribution of the exoplanet pipeline to the Gaia DR3 sample of astrometric orbital solutions by describing the methods used for fitting the orbits, the identification of significant solutions, and their validation. We then present an overview of the statistical properties of the solution parameters. Methods. Using both a Markov chain Monte Carlo and a genetic algorithm, we fitted the 34 months of Gaia DR3 astrometric time series with a single Keplerian astrometric-orbit model that had 12 free parameters and an additional jitter term, and retained the solutions with the lowest χ 2 . Verification and validation steps were taken using significance tests, internal consistency checks using the Gaia radial velocity measurements (when available), as well as literature radial velocity and astrometric data, leading to a subset of candidates that were labelled “validated”. Results. We determined astrometric-orbit solutions for 1162 sources, and 198 solutions were assigned the “Validated” label. Precise companion-mass estimates require external information and are presented elsewhere. To broadly categorise the different mass regimes in this paper, we use the pseudo-companion mass M̃ c assuming a solar-mass host and define three solution groups: 17 (9 validated) solutions with companions in the planetary-mass regime ( M̃ c < 20 M J ), 52 (29 validated) in the brown dwarf regime (20 M J ≤ M̃ c ≤ 120 M J ), and 1093 (160 validated) in the low-mass stellar companion regime ( M̃ c > 120 M J ). From internal and external verification and validation, we estimate the level of spurious and incorrect solutions in our sample to be ∼5% and ∼10% in the ‘OrbitalAlternative’ and ‘OrbitalTargetedSearch’ candidate sample, respectively. Conclusions. We demonstrate that Gaia is able to confirm and sometimes refine the orbits of known orbital companions and to identify new candidates, providing us with a positive outlook for the expected harvest from the full mission data in future data releases.