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The hunt for compact objects is on. Rarely seen massive binaries with a compact object are a crucial phase in the evolution towards compact object mergers. In Gaia data release 3 (DR3), the first Gaia astrometric orbital solutions for binary sources will become available. We investigate how many black holes (BH) with massive main-sequence dwarf companions (OB+BH binaries) are expected to be detected as binaries in DR3 and at the end of the nominal 5-yr mission (DR4). We estimate the fraction of identifiable OB+BH binaries and discuss the distributions of the masses of both components and the orbital periods. We study the impact of different BH-formation scenarios. Using tailored models for the massive star population, which assume a direct collapse and no kick upon BH formation (the fiducial case), we estimate the fraction of OB+BH systems that Gaia will detect as binaries. A distance distribution according to that of the second Alma Luminous Star catalogue (ALSII) is assumed. We investigate how many of the systems detected as binaries are identifiable as OB+BH binaries, using a method based on astrometric data. In the fiducial case we conservatively estimate that 77% of the OB+BH binaries in ALSII will be detected as binaries in DR3, of which 89% are identifiable as OB+BH binaries. This leads to a total of around 190 OB+BH binaries, a 20-fold increase in the known sample of OB+BH binaries, covering an uncharted parameter space of long-period binaries. The size and properties of the identifiable OB+BH population will contain crucial observational constraints to improve our understanding of BH formation. In DR4, the detected fraction will increase to 85%, of which 82% will be identifiable. Hence, an additional ~5 systems could be identified, which are expected to have either very short or long periods. The fractions become smaller for different BH-formation scenarios. (truncated), Comment: 22 pages, 18 figures, 2 tables, accepted for publication in A&A