We search for hints to the origin and nature of compact stellar systems in the magnitude range of ultracompact dwarf galaxies in deep wide-field imaging data of the Fornax cluster core. We visually investigate a large sample of 355 spectroscopically confirmed cluster members with V-band equivalent magnitudes brighter than -10mag for faint extended structures. Our data reveal peculiar compact stellar systems, which appear asymmetric or elongated from their outer light distribution. We characterize the structure of our objects by quantifying their core concentration, as well as their outer asymmetry and ellipticity. For the brighter objects of our sample we also investigate their spatial and phase-space distribution within the cluster. We argue that the distorted outer structure alone that is seen for some of our objects, is not sufficient to decide whether these systems have a star cluster or a galaxy origin. However, we find that objects with low core concentration and high asymmetry (or high ellipticity) are primarily located at larger cluster-centric distances as compared to the entire sample. This supports the hypothesis that at least some of these objects may originate from tidally stripped galaxies.