Permissioned blockchains are a promising technology for secure decentralized data management in business-to-business use-cases. In contrast to Bitcoin and similar public blockchains, which rely on Proof-of-Work for consensus and are deployed on thousands of geo-distributed nodes, business-to-business use-cases (such as supply chain management and banking) require significantly fewer nodes, cheaper consensus, and are often deployed in controlled environments with fast networking and low latency. However, permissioned blockchains often follow the architectural thinking behind their WAN-oriented public relatives, which results in end-to-end latency several orders of magnitude higher than necessary. In this work, we propose a fundamental shift in permissioned blockchain design, eliminating blocks in favor of processing transactions in a streaming fashion. This results in a drastically lower latency without reducing throughput or forfeiting reliability and security guarantees. To demonstrate the wide applicability of our design, we prototype StreamChain based on the Hyperledger Fabric, and show that it delivers latency two orders of magnitude lower than Fabric, while sustaining similar throughput. This performance makes StreamChain a potential alternative to traditional databases and, thanks to its streaming paradigm, enables further research, for instance, around reducing latency through relying on modern programmable hardware accelerators.