Semiconductors have already had a very profound effect on society, accelerating scientific research and driving greater connectivity. Future semiconductor hardware will open up new possibilities in quantum computing, artificial intelligence and edge computing, for applications such as cybersecurity and personalized healthcare. By nature of its ethos, open hardware provides opportunities for even greater collaboration and innovations across education, academic research and industry. Here we present Flex-RV, a 32-bit microprocessor based on an open RISC-V instruction set fabricated with indium gallium zinc oxide thin-film transistors on a flexible polyimide substrate, enabling an ultralow-cost bendable microprocessor. Flex-RV also integrates a programmable machine learning (ML) hardware accelerator inside the microprocessor and demonstrates new instructions to extend the RISC-V instruction set to run ML workloads. It is implemented, fabricated and demonstrated to operate at 60 kHz consuming less than 6 mW power. Its functionality when assembled onto a flexible printed circuit board is validated while executing programs under flat and tight bending conditions, achieving no worse than 4.3% performance variation on average. Flex-RV pioneers an era of sub-dollar open standard non-silicon 32-bit microprocessors and will democratize access to computing and unlock emerging applications in wearables, healthcare devices and smart packaging.