Flexible robotic fish are emerging as a promising class of deep-sea exploration platforms because they combine compliant bodies, low-disturbance fish-like propulsion, and the potential for distributed sensing and autonomy. Unlike conventional biomimetic robotic fish developed mainly for shallow or moderate-depth environments, deep-sea flexible robotic fish must simultaneously address high hydrostatic pressure, low temperature, darkness, limited communication, constrained power supply, and complex near-bottom terrain. This review synthesizes research at the intersection of deep-sea soft robotics, bio-inspired robotic fish, smart-material actuation, pressure-adaptive packaging, multimodal sensing, and autonomous control. The literature is organized around a system-level design chain: biological mechanisms that inspire pressure adaptation and perception, body architectures that distribute pressure and protect electronics, soft actuators that generate fish-like propulsion, and control strategies that enable near-bottom and long-duration tasks. The review highlights that the central challenge is not any single actuator or material, but the co-design of pressure-adaptive bodies, hybrid soft actuation, reliable interfaces, multimodal perception, energy management, and autonomy. To strengthen engineering translation, this revised review further adds design-principle abstraction, actuator-selection guidance, prototype-level comparison, failure-mode analysis, and a computational design workflow. Future research should prioritize long-term reliability tests, standardized deep-sea evaluation protocols, physics-informed modeling, and integrated prototype demonstrations under realistic mission conditions.