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Context. As part of the third Gaia data release, we present the contributions of the non-stellar and classification modules from the eighth coordination unit (CU8) of the Data Processing and Analysis Consortium, which is responsible for the determination of source astrophysical parameters using Gaia data. This is the third in a series of three papers describing the work done within CU8 for this release. Aims. For each of the five relevant modules from CU8, we summarise their objectives, the methods they employ, their performance, and the results they produce for Gaia DR3. We further advise how to use these data products and highlight some limitations. Methods. The Discrete Source Classifier (DSC) module provides classification probabilities associated with five types of sources: quasars, galaxies, stars, white dwarfs, and physical binary stars. A subset of these sources are processed by the Outlier Analysis (OA) module, which performs an unsupervised clustering analysis, and then associates labels with the clusters to complement the DSC classification. The Quasi Stellar Object Classifier (QSOC) and the Unresolved Galaxy Classifier (UGC) determine the redshifts of the sources classified as quasar and galaxy by the DSC module. Finally, the Total Galactic Extinction (TGE) module uses the extinctions of individual stars determined by another CU8 module to determine the asymptotic extinction along all lines of sight for Galactic latitudes |b| > 5 deg. Results. Gaia DR3 includes 1591 million sources with DSC classifications; 56 million sources to which the OA clustering is applied; 1.4 million sources with redshift estimates from UGC; 6.4 million sources with QSOC redshift; and 3.1 million level 9 HEALPixes of size 0.013 squared degree, where the extinction is evaluated by TGE.