Despite being a common figure of speech, hyperbole is under-researched in Figurative Language Processing. In this paper, we tackle the challenging task of hyperbole generation to transfer a literal sentence into its hyperbolic paraphrase. To address the lack of available hyperbolic sentences, we construct HYPO-XL, the first large-scale English hyperbole corpus containing 17,862 hyperbolic sentences in a non-trivial way. Based on our corpus, we propose an unsupervised method for hyperbole generation that does not require parallel literalhyperbole pairs. During training, we fine-tune BART (Lewis et al., 2020) to infill masked hyperbolic spans of sentences from HYPO-XL. During inference, we mask part of an input literal sentence and over-generate multiple possible hyperbolic versions. Then a BERT-based ranker selects the best candidate by hyperbolicity and paraphrase quality. Automatic and human evaluation results show that our model is effective at generating hyperbolic paraphrase sentences and outperforms several baseline systems.