All existing databases of spoofed speech contain attack data that is spoofed in its entirety. In practice, it is entirely plausible that successful attacks can be mounted with utterances that are only partially spoofed. By definition, partially-spoofed utterances contain a mix of both spoofed and bona fide segments, which will likely degrade the performance of countermeasures trained with entirely spoofed utterances. This hypothesis raises the obvious question: 'Can we detect partially-spoofed audio?' This paper introduces a new database of partially-spoofed data, named PartialSpoof, to help address this question. This new database enables us to investigate and compare the performance of countermeasures on both utterance- and segmental- level labels. Experimental results using the utterance-level labels reveal that the reliability of countermeasures trained to detect fully-spoofed data is found to degrade substantially when tested with partially-spoofed data, whereas training on partially-spoofed data performs reliably in the case of both fully- and partially-spoofed utterances. Additional experiments using segmental-level labels show that spotting injected spoofed segments included in an utterance is a much more challenging task even if the latest countermeasure models are used.