Manta rays use wing-like pectoral fins for intriguing oscillatory swimming. It provides rich inspiration for designing potentially fast, efficient, and maneuverable soft swimming robots, which, however, have yet to be realized. It remains a grand challenge to combine fast speed, high efficiency, and high maneuverability in a single soft swimmer while using simple actuation and control. Here, we report leveraging spontaneous snapping stroke in the monostable flapping wing of a manta-like soft swimmer to address the challenge. The monostable wing is pneumatically actuated to instantaneously snap through to stroke down, and upon deflation, it will spontaneously stroke up by snapping back to its initial state, driven by elastic restoring force, without consuming additional energy. This largely simplifies designs, actuation, and control for achieving a record-high speed of 6.8 body length per second, high energy efficiency, and high maneuverability and collision resilience in navigating through underwater unstructured environments with obstacles by simply tuning single-input actuation frequencies.