NVIDIA Multi-Process Service (MPS) enables fine-grained GPU sharing by allowing multiple processes to execute concurrently on the same GPU, making it an important mechanism for improving GPU utilization. However, MPS has weak fault resilience: a fault in one process can terminate all co-running processes, limiting its adoption in resilience-critical settings such as multi-tenant GPU clusters. In this work, we design fault-resilient MPS to solve this problem. Our design is guided by insights from a systematic characterization of GPU faults and a deep analysis of their end-to-end processing pipeline. Based on these insights, we design two complementary mechanisms. A fault isolation mechanism for the dominant memory-related faults that can be fully isolated by software intervention in the open GPU driver kernel module. For other faults whose process is within proprietary software, we design a practical mechanism -- fast recovery using virtual memory based GPU-resident state sharing. Our evaluation on different GPUs and workloads shows that these mechanisms can handle corresponding faults effectively with minimal overhead.