In the field of marine biology-inspired robotics, anguilliform locomotion, as one of the most common underwater propulsion types, has been widely studied and implemented in many robot prototypes. Most of these robots consisted of rigid parts, and they were able to generate smooth sine waves along the robot body using a great number of rigid segments, electric motors, and complicated control. To simplify the robot structures and improve body compliance, anguilliform robots with a full soft body are highly desirable for better biomimicry. In this article, we propose a serial soft-actuator array consisting of four fiber-reinforced, bidirectionally bending, fluidic elastomer actuators (FEAs) to achieve the generation of anguilliform body waves. The FEA is fabricated using the typical soft lithography method with dual-chamber configuration and fiber reinforcement. The bending performance of the single FEA in free space with static loads was measured and compared with the three-dimensional simulation results using the finite element method. After that, the FEA array was assembled using the through-chamber tube connection and tested in water to investigate the bending performance with dynamic loads. Finally, with coordinated control of each segment, this FEA array achieved the generation of anguilliform body waves with different frequencies and amplitudes, which unveils a novel and promising approach to develop an anguilliform underwater robot. By attaching this FEA array to a miniaturized stand-alone station for control and actuation, an untethered swimming robot was built and successfully propelled by the FEA array, which demonstrates the capabilities and potentials of this soft and slender robot body design.