In recent years, GNSS-based cooperative positioning applications such as multiple unmanned aerial vehicles (UAVs), multiple unmanned surface vehicles (USVs), and intelligent transportation systems (ITSs) are receiving a great attention. These applications can achieve the information exchange among wireless devices (nodes). As multiple devices equipped with GNSS receivers are feasible to measure the distance and communicate with each other, the positioning performance of a single device can be enhanced greatly by using redundant measurements. Up to now, the focus of GNSS-based cooperative positioning applications is to provide users with high positioning accuracy and availability. Meanwhile, there is also an increasing demand for the integrity of GNSS-based cooperative positioning applications. Compared with conventional GNSS applications, cooperative positioning applications are more vulnerable to the interferences or attacks from the ground. For a cooperative positioning system, the faults occurring at one device could also affect other devices, further resulting in severe positioning errors of each node. Receiver autonomous integrity monitoring (RAIM), as a prevalent integrity monitoring method, has shown good performance on detecting and excluding the faults related with satellites. However, some challenges have to be tackled for applying RAIM to GNSS-based cooperative positioning applications including the definition, detection, and exclusion of new fault modes. Therefore, it is necessary to analyze the potential faults modes and design an effective RAIM algorithm for cooperative positioning system. In our research, the fault modes related with nodes are firstly introduced. A whole-network extended Kalman filter (WNEKF) is then proposed to integrate the pseudorange measurements of satellites and the ranging measurements between the nodes. Based on solution separation strategy, an integrity monitoring algorithm is developed to detect and exclude the faults. The benefit of our method is that it is conductive to monitor all the faults of the whole network simultaneously. Besides, based on the centralized algorithm, only few master nodes are needed to execute the integrity monitoring algorithm, while the other nodes receive integrity information from the master nodes.