Data for building fact-checking models for Italian is scarce, often contains ambiguous claims, and lacks textual diversity. This makes it hard to reliably apply such tools in the real world to support fact-checkers’ work. In this paper, we propose a categorization of claim ambiguity and label the largest Italian test set based on it. Moreover, we create challenge sets across two axes of variation: genres and fact-checking sources. Our experiments using transformer-based semantic search show a large drop in performance under domain shift, and indicate the benefit of models’ abstention in case of lacking evidence.