Fire extinguishers are a necessary component of crewed space missions, providing a measure of fire safety beyond material selection and other controls. The elevated oxygen levels and presence of reduced but significant buoyancy during proposed Lunar and Martian missions presents both elevated fire risk and challenges in understanding and designing for extinguisher performance. This white paper proposes experimental and modeling research into the performance of fire extinguishers in Lunar and Martian atmosphere and gravity environments, exploring the use of additives to water extinguishers and research into alternative extinguishing materials, and exploration into the use of Froude number scaling relationships, used in terrestrial water extinguishing research and design, in the Lunar and Martian environment.