This paper is about few-shot instance segmentation, where training and test image sets do not share the same object classes. We specify and evaluate a new few-shot anchor-free part-based instance segmenter FAPIS. Our key novelty is in explicit modeling of latent object parts shared across training object classes, which is expected to facilitate our few-shot learning on new classes in testing. We specify a new anchor-free object detector aimed at scoring and regressing locations of foreground bounding boxes, as well as estimating relative importance of latent parts within each box. Also, we specify a new network for delineating and weighting latent parts for the final instance segmentation within every detected bounding box. Our evaluation on the benchmark COCO-20i dataset demonstrates that we significantly outperform the state of the art.