This paper presents Banyan, the first rotating leader state machine replication (SMR) protocol that allows transactions to be confirmed in just a single round-trip time in the Byzantine fault tolerance (BFT) setting. Based on minimal alterations to the Internet Computer Consensus (ICC) protocol and with negligible communication overhead, we introduce a novel dual mode mechanism that enables optimal block finalization latency in the fast path. Crucially, the modes of operation are integrated, such that even if the fast path is not effective, no penalties are incurred. Moreover, our algorithm maintains the core attributes of the ICC protocol it is based on, including optimistic responsiveness and rotating leaders without the necessity for a view-change protocol. We prove the correctness of our protocol and provide an open-source implementation of it. Banyan is compared to its predecessor ICC, as well as other well known BFT protocols, in a globally distributed wide-area network. Our evaluation reveals that Banyan reduces latency by up to 30% compared to state-of-the-art protocols, without requiring additional security assumptions.