The deep sea remains the largest unknown territory on Earth because it is so difficult to explore1-4. Owing to the extremely high pressure in the deep sea, rigid vessels5-7 and pressure-compensation systems8-10 are typically required to protect mechatronic systems. However, deep-sea creatures that lack bulky or heavy pressure-tolerant systems can thrive at extreme depths11-17. Here, inspired by the structure of a deep-sea snailfish15, we develop an untethered soft robot for deep-sea exploration, with onboard power, control and actuation protected from pressure by integrating electronics in a silicone matrix. This self-powered robot eliminates the requirement for any rigid vessel. To reduce shear stress at the interfaces between electronic components, we decentralize the electronics by increasing the distance between components or separating them from the printed circuit board. Careful design of the dielectric elastomer material used for the robot's flapping fins allowed the robot to be actuated successfully in a field test in the Mariana Trench down to a depth of 10,900 metres and to swim freely in the South China Sea at a depth of 3,224 metres. We validate the pressure resilience of the electronic components and soft actuators through systematic experiments and theoretical analyses. Our work highlights the potential of designing soft, lightweight devices for use in extreme conditions.