The PSI Positron Production project (P³ or P-cubed) is a demonstrator for a novel positron source for FCC-ee. The high current requirements of future colliders can be compromised by the extremely high positron emittance at the production target and consequent poor capture and transport to the damping ring. However, recent advances in high-temperature superconductors allow for a highly efficient matching of such an emittance through the use a solenoid around the target delivering a field over 10 T on-axis. Moreover, the emittance of the matched positron beam can be contained through large aperture RF cavities surrounded by a multi-Tesla field generated by conventional superconducting solenoids, where simulations estimate a yield higher by one order of magnitude with respect to the state-of-the-art. The goal of P³ is to demonstrate this basic principle by implementing the aforementioned solenoids into a prototype positron source based on a 6 GeV electron beam from the SwissFEL linac, two RF capture cavities and a beam diagnostics section.