Peer-to-peer (P2P) energy trading among neighbouring prosumers is considered as a promising trading method for the future smart grid. Demand response management becomes a critical challenge due to increased penetration of renewable energy. Earlier work mostly considers P2P trading models with only prosumers, while we believe there will still be a role to play for electricity suppliers in local energy markets in the foreseeable future. This paper therefore proposes a trading model in a community based P2P electric energy market that includes local energy suppliers and a community coordinator as market participants, in addition to pure energy consumers, and prosumers. We develop a demand response mechanism for the proposed trading model, in which dynamic pricing for suppliers is used. The community coordinator negotiates with suppliers on the external energy price and trades with them on behalf of the households within the P2P energy market. In our proposed trading model, the behaviour of the suppliers and the community households are modelled as two non-cooperative games. We propose a distributed algorithm to determine the equilibrium of the games. Simulation results show that our model has great effect on reducing the net peak load and increasing the market participants’ profit. Additionally, the proposed mechanism is shown to act as an efficient incentive for pure energy consumers to become prosumers.