Recently, deep generative models have been regarded as promising tools in fragment-based drug design (FBDD). Despite the growing interest in these models, they still face challenges in generating molecules with desired properties in low data regimes. In this study, we propose a novel flow-based autoregressive model named FFLOM for linker and R-group design. In a large-scale benchmark evaluation on ZINC, CASF, and PDBbind test sets, FFLOM achieves state-of-the-art performance in terms of validity, uniqueness, novelty, and recovery of the generated molecules and can recover over 92% of the original molecules in the PDBbind test set (with at least five atoms). FFLOM also exhibits excellent potential applicability in several practical scenarios encompassing fragment linking, PROTAC design, R-group growing, and R-group optimization. In all four cases, FFLOM can perfectly reconstruct the ground-truth compounds and generate over 74% of molecules with novel fragments, some of which have higher binding affinity than the ground truth.