Distributed prefix caching accelerates long-context LLM serving by reusing KV cache entries for common context prefixes. However, KV cache fetches can become a bottleneck when network bandwidth is limited. Compression mitigates the bandwidth issue, but can degrade overall performance when decompression interferes with model computation. We present ShadowServe, the first SmartNIC-accelerated, interference-free prefix caching system for LLM serving. ShadowServe separates a control plane on the host and a data plane fully offloaded to the SmartNIC, which eliminates interference to both host GPU and CPU. To overcome the SmartNIC's limited compute and memory resources, we design a chunked pipeline that parallelizes data plane operations across the SmartNIC's compute resources, and a minimal-copy memory management scheme that reduces memory pressure on the SmartNIC. Compared to state-of-the-art solutions, ShadowServe achieves up to 2.2x lower loaded time-per-output-token (TPOT), and reduces time-to-first-token (TTFT) by up to 1.38x in low-bandwidth scenarios (<= 20 Gbps), translating to up to 1.35x higher throughput.