This paper presents the Dravidian Offensive Span Identification Dataset (DOSA) for under-resourced Tamil-English and Kannada-English code-mixed text. The dataset addresses the lack of code-mixed datasets with annotated offensive spans by extending annotations of existing code-mixed offensive language identification datasets. It provides span annotations for Tamil-English and Kannada-English code-mixed comments posted by users on YouTube social media. Overall the dataset consists of 4786 Tamil-English comments with 6202 annotated spans and 1097 Kannada-English comments with 1641 annotated spans, each annotated by two different annotators. We further present some of our baseline experimental results on the developed dataset, thereby eliciting research in under-resourced languages, leading to an essential step towards semi-automated content moderation in Dravidian languages. The dataset is available in https://github.com/teamdl-mlsg/DOSA