The rigid MPI programming model and batch scheduling dominate high-performance computing. While clouds brought new levels of elasticity into the world of computing, supercomputers still suffer from low resource utilization rates. To enhance supercomputing clusters with the benefits of serverless computing, a modern cloud programming paradigm for pay-as-you-go execution of stateless functions, we present rFaaS, the first RDMA-aware Function-as-a-Service (FaaS) platform. With hot invocations and decentralized function placement, we overcome the major performance limitations of FaaS systems and provide low-latency remote invocations in multi-tenant environments. We evaluate the new serverless system through a series of microbenchmarks and show that remote functions execute with negligible performance overheads. We demonstrate how serverless computing can bring elastic resource management into MPI-based high-performance applications. Overall, our results show that MPI applications can benefit from modern cloud programming paradigms to guarantee high performance at lower resource costs.