N-ary facts composed of a primary triple (head entity, relation, tail entity) and an arbitrary number of auxiliary attribute-value pairs, are prevalent in real-world knowledge graphs (KGs). Link prediction on n-ary facts is to predict a missing element in an n-ary fact. This helps populate and enrich KGs and further promotes numerous downstream applications. Previous studies usually require a substantial amount of high-quality data to understand the elements in n-ary facts. However, these studies overlook few-shot relations, which have limited labeled instances, yet are common in real-world scenarios. Thus, this paper introduces a new task, few-shot link prediction on n-ary facts. It aims to predict a missing entity in an n-ary fact with limited labeled instances. We further propose a model for Few-shot Link prEdict on N-ary facts, thus called FLEN, which consists of three modules: the relation learning, support-specific adjusting, and query inference modules. FLEN captures relation meta information from limited instances to predict a missing entity in a query instance. To validate the effectiveness of FLEN, we construct three datasets based on existing benchmark data. Our experimental results show that FLEN significantly outperforms existing related models in both few-shot link prediction on n-ary facts and binary facts.