This paper proposes a new local energy market design for distribution systems, which integrates peer-to-peer (P2P) energy trading and probabilistic locational marginal pricing. Distribution locational marginal pricing and P2P energy trading have each been proposed as potential alternatives to traditional retail pricing, to improve coordination between prosumers with distributed energy resources. Unidirectional locational pricing provides a scalable approach for coordinating demand, considering constraints and losses; while P2P energy trading allows prosumers to negotiate mutually beneficial bilateral energy transactions that increase the utilisation of their flexible energy resources. This paper proposes a market design combining the benefits of these two strategies. First, a new strategy for dayahead locational marginal pricing is developed, which manages the uncertainty associated with local generation, demand and upstream prices by introducing a spread between the prices charged for energy imports and paid for energy exports. Then, local P2P energy trading platforms are integrated to additionally enable direct prosumer-to-prosumer trading, with transaction fees penalising energy transfers according to probabilistic differential locational marginal prices. Case studies are presented for a multi-phase low voltage distribution network, showing how the design can create value for prosumers, and the system as a whole, by reducing the curtailment of renewable generation.