Superconducting computing promises enhanced computational power in both classical and quantum approaches. Yet, efficient schemes for scalable and fast superconducting memories are still missing. On the one hand, the large inductance required in magnetic flux-controlled Josephson memories impedes device miniaturization and scalability. On the other hand, schemes based on the ferromagnetic order to store information often degrades superconductivity, and limits the operation speed to the magnetization switching rate of a few GHz. Here, we overcome these limitations with a fully superconducting memory cell based on the hysteretic phase-slip transition existing in long aluminum nanowire Josephson junctions. The memory logic state is codified in the topological index of the junction providing a robust protection against stocastic phase slips and magnetic flux noise. Our direct and non-destructive read-out schemes, based on local DC or AC tunneling spectroscopy, ensure reduced dissipation (≤ 40 fW) thereby yielding a very low energy per bit read-out power consumption as low as ~ 10-24 J as estimated from the typical time response of the structure (≤ 30 ps). The memory, measured over several days, showed no evidence of information degradation up to ~1.1 K, i.e., ~85% of the critical temperature of aluminum. The ease of operation combined with remarkable performance elects the Josephson phase-slip memory as an attractive storage cell to be exploited in advanced superconducting classical logic architectures or flux qubits.