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We describe the fast transient classification algorithm in the center of the kilonova (KN) science module currently implemented in the Fink broker and report classification results based on simulated catalogs and real data from the ZTF alert stream. We used noiseless, homogeneously sampled simulations to construct a basis of principal components (PCs). All light curves from a more realistic ZTF simulation were written as a linear combination of this basis. The corresponding coefficients were used as features in training a random forest classifier. The same method was applied to long (>30 days) and medium (<30 days) light curves. The latter aimed to simulate the data situation found within the ZTF alert stream. Classification based on long light curves achieved 73.87% precision and 82.19% recall. Medium baseline analysis resulted in 69.30% precision and 69.74% recall, thus confirming the robustness of precision results when limited to 30 days of observations. In both cases, dwarf flares and point Type Ia supernovae were the most frequent contaminants. The final trained model was integrated into the Fink broker and has been distributing fast transients, tagged as KN_candidates, to the astronomical community, especially through the GRANDMA collaboration. We showed that features specifically designed to grasp different light curve behaviors provide enough information to separate fast (KN-like) from slow (non-KN-like) evolving events. This module represents one crucial link in an intricate chain of infrastructure elements for multi-messenger astronomy which is currently being put in place by the Fink broker team in preparation for the arrival of data from the Vera Rubin Observatory Legacy Survey of Space and Time.