A fuel supply system having a low pressure region, a pumping device to deliver fuel from the low pressure region to a high pressure region. In the high pressure region between the pumping device and injectors there is a pressure storage system that is permanently under high pressure. The pressure storage system has a plurality of distributor units each with at least three connections connected in series. A respective injector connection of each distributor unit is connected to at least one injector each via a high pressure line that is under high pressure at times dependent on the injection cycle. Each distributor unit of the pressure storage system is assigned an individual leakage detection device. Each distributor unit is assigned a non-return valve, which allows a leakage flow starting out from the respective distributor unit in the direction of the pumping device.