Reasoning is a hallmark of human intelligence, enabling adaptive decision-making in complex and unfamiliar scenarios. In contrast, machine intelligence remains bound to training data, lacking the ability to dynamically refine solutions at inference time. While some recent advances have explored reasoning in machines, these efforts are largely limited to verbal domains such as mathematical problem-solving, where explicit rules govern step-by-step reasoning. Other critical real-world tasks - including visual perception, spatial reasoning, and radiological diagnosis - require non-verbal reasoning, which remains an open challenge. Here we present a novel learning paradigm that enables machine reasoning in vision by allowing performance improvement with increasing thinking time (inference-time compute), even under conditions where labelled data is very limited. Inspired by dual-process theories of human cognition in psychology, our approach integrates a fast-thinking System I module for familiar tasks, with a slow-thinking System II module that iteratively refines solutions using self-play reinforcement learning. This paradigm mimics human reasoning by proposing, competing over, and refining solutions in data-scarce scenarios. We demonstrate superior performance through extended thinking time, compared not only to large-scale supervised learning but also foundation models and even human experts, in real-world vision tasks. These tasks include computer-vision benchmarks and cancer localisation on medical images across five organs, showcasing transformative potential for non-verbal machine reasoning.