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We present a catalog of quasars and corresponding redshifts in the Kilo-Degree Survey (KiDS) Data Release 4. We trained machine learning (ML) models, using optical ugri and near-infrared ZYJHK_s bands, on objects known from Sloan Digital Sky Survey (SDSS) spectroscopy. We define inference subsets from the 45 million objects of the KiDS photometric data limited to 9-band detections. We show that projections of the high-dimensional feature space can be successfully used to investigate the estimations. The model creation employs two test subsets: randomly selected and the faintest objects, which allows to fit the bias versus variance trade-off. We tested three ML models: random forest (RF), XGBoost (XGB), and artificial neural network (ANN). We find that XGB is the most robust model for classification, while ANN performs the best for combined classification and redshift. The inference results are tested using number counts, Gaia parallaxes, and other quasar catalogs. Based on these tests, we derived the minimum classification probability which provides the best purity versus completeness trade-off: p(QSO_cand) > 0.9 for r < 22 and p(QSO_cand) > 0.98 for 22 < r < 23.5. We find 158,000 quasar candidates in the safe inference subset (r < 22) and an additional 185,000 candidates in the reliable extrapolation regime (22 < r < 23.5). Test-data purity equals 97% and completeness is 94%; the latter drops by 3% in the extrapolation to data fainter by one magnitude than the training set. The photometric redshifts were modeled with Gaussian uncertainties. The redshift error (mean and scatter) equals 0.01 +/- 0.1 in the safe subset and -0.0004 +/- 0.2 in the extrapolation, in a redshift range of 0.14 < z < 3.63. Our success of the extrapolation challenges the way that models are optimized and applied at the faint data end. The catalog is ready for cosmology and active galactic nucleus (AGN) studies.