This paper studies the real-time energy trading problem in a smart community consisting of a group of grid-connected prosumers with controllable loads, renewable generations and energy storage systems. We propose a peer-to-peer (P2P) energy trading system, which integrates energy trading with energy management, enabling each prosumer to jointly manage its energy consumption, storage scheduling and energy trading in a dynamic manner for smart communities consisting of a group of grid-connected prosumers with controllable loads, renewable generations and energy storage systems. The proposed community-based P2P energy trading system combines an online energy control and trading algorithm with a double auction mechanism. The energy control and trading algorithm is designed based on the Lyapunov theory, allowing each prosumer to independently determine its bid in each time slot only based on its current energy supply condition, while the trading price, which is determined via the double auction mechanism, reflects the collective energy supply conditions of all prosumers participating in energy trading. The integration of the Lyapunov-based energy control and trading algorithm and the double auction mechanism yields a dynamic energy trading pricing mechanism that induces the prosumers to participate in energy trading in a coordinated manner by influencing the energy consumption, energy charging/discharging and energy trading decisions of the prosumers. Numerical simulation results demonstrate that energy exchange in the proposed scalable energy trading system yields significant improvements in terms of energy cost savings and renewable energy utilization efficiency, while ensuring the fair sharing of the benefits reaped from energy trading among the prosumers.