Recently, Large language models (LLMs) with in-context learning have demonstrated remarkable potential in handling neural machine translation. However, existing evidence shows that LLMs are prompt-sensitive and it is sub-optimal to apply the fixed prompt to any input for downstream machine translation tasks. To address this issue, we propose an adaptive few-shot prompting (AFSP) framework to automatically select suitable translation demonstrations for various source input sentences to further elicit the translation capability of an LLM for better machine translation. First, we build a translation demonstration retrieval module based on LLM's embedding to retrieve top-k semantic-similar translation demonstrations from aligned parallel translation corpus. Rather than using other embedding models for semantic demonstration retrieval, we build a hybrid demonstration retrieval module based on the embedding layer of the deployed LLM to build better input representation for retrieving more semantic-related translation demonstrations. Then, to ensure better semantic consistency between source inputs and target outputs, we force the deployed LLM itself to generate multiple output candidates in the target language with the help of translation demonstrations and rerank these candidates. Besides, to better evaluate the effectiveness of our AFSP framework on the latest language and extend the research boundary of neural machine translation, we construct a high-quality diplomatic Chinese-English parallel dataset that consists of 5,528 parallel Chinese-English sentences. Finally, extensive experiments on the proposed diplomatic Chinese-English parallel dataset and the United Nations Parallel Corpus (Chinese-English part) show the effectiveness and superiority of our proposed AFSP.