Energy storage capacity leasing is becoming a new business model. It has received a lot of attention because of its low risk and high flexibility. To sovle the problem of energy storage capacity leasing between an independent energy storage operator and multiple prosumers who can simultaneously produce and consume energy in an energy community under asymmetric cost information, first, asymmetric information is transformed into complete information via the Harsanyi transformation. Second, a two-stage extensive game model between the energy storage operator and the prosumers is established. Third, market clearing and settlement rules based on a uniform price mechanism are designed. Finally, the subgame refinement Nash equilibrium of the extensive game with complete information between the operator and the prosumers under the uniform price mechanism is analyzed, and the existence of the equilibrium is proved by the Brouwer fixed point theorem. Simulation studies show that energy storage capacity leasing can significantly reduce the electricity cost of the prosumers and improve the efficiency of energy storage utilization. The game equilibrium of the uniform price mechanism is superior to that of the discriminatory price mechanism in finding fair energy storage capacity leasing prices, increasing the profits of the independent energy storage operator, increasing the scale of energy storage capacity transactions, and improving social welfare.