Automated fact-checking systems verify claims against evidence to predict their veracity. In real-world scenarios, the retrieved evidence may not unambiguously support or refute the claim and yield conflicting but valid interpretations. Existing fact-checking datasets assume that the models developed with them predict a single veracity label for each claim, thus discouraging the handling of such ambiguity. To address this issue we present AmbiFC , 1 a fact-checking dataset with 10k claims derived from real-world information needs. It contains fine-grained evidence annotations of 50k passages from 5k Wikipedia pages. We analyze the disagreements arising from ambiguity when comparing claims against evidence in AmbiFC , observing a strong correlation of annotator disagreement with linguistic phenomena such as underspecification and probabilistic reasoning. We develop models for predicting veracity handling this ambiguity via soft labels, and find that a pipeline that learns the label distribution for sentence-level evidence selection and veracity prediction yields the best performance. We compare models trained on different subsets of AmbiFC and show that models trained on the ambiguous instances perform better when faced with the identified linguistic phenomena.