African languages have far less in-language content available digitally, making it challenging for question answering systems to satisfy the information needs of users. Cross-lingual open-retrieval question answering (XOR QA) systems -- those that retrieve answer content from other languages while serving people in their native language -- offer a means of filling this gap. To this end, we create AfriQA, the first cross-lingual QA dataset with a focus on African languages. AfriQA includes 12,000+ XOR QA examples across 10 African languages. While previous datasets have focused primarily on languages where cross-lingual QA augments coverage from the target language, AfriQA focuses on languages where cross-lingual answer content is the only high-coverage source of answer content. Because of this, we argue that African languages are one of the most important and realistic use cases for XOR QA. Our experiments demonstrate the poor performance of automatic translation and multilingual retrieval methods. Overall, AfriQA proves challenging for state-of-the-art QA models. We hope that the dataset enables the development of more equitable QA technology.