Federated learning (FL) is a machine learning framework for collaborative data mining in many scenarios (e.g. Internet of Things) due to its privacy-preserving feature. However, various attacks arise security concerns of FL, such as poisoning, backdoor, and DDoS attacks. Several blockchain-based FL schemes strengthen credibility and security without considering the increased communication overhead. Some existing work compresses local updated gradients to sign vectors to lower communication overhead at the expense of model accuracy. To address the above concerns, this paper offers a blockchain-based asynchronous SignSGD (BASS) scheme. A novel asynchronous sign aggregation algorithm is introduced to ensure model accuracy even if the local updated gradients are compressed to sign vectors. Considering the unstable network connection on IoT, a consensus algorithm that elects multiple leader nodes enables reliable global model aggregation. The introduced blockchain improves credibility and security without downgrading efficiency. Empirical studies show that BASS outperforms other schemes in efficiency, model accuracy, and security.