Few-shot continual relation extraction aims to continually train a model on incrementally few-shot data to learn new relations while avoiding forgetting old ones. However, current memory-based methods are prone to overfitting memory samples, resulting in insufficient activation of old relations and limited ability to handle the confusion of similar classes. In this paper, we design a new N-way-K-shot Continual Relation Extraction (NK-CRE) task and propose a novel few-shot continual relation extraction method with Consistent Prototype Learning (ConPL) to address the aforementioned issues. Our proposed ConPL is mainly composed of three modules: 1) a prototype-based classification module that provides primary relation predictions under few-shot continual learning; 2) a memory-enhanced module designed to select vital samples and refined prototypical representations as a novel multi-information episodic memory; 3) a consistent learning module to reduce catastrophic forgetting by enforcing distribution consistency. To effectively mitigate catastrophic forgetting, ConPL ensures that the samples and prototypes in the episodic memory remain consistent in terms of classification and distribution. Additionally, ConPL uses prompt learning to extract better representations and adopts a focal loss to alleviate the confusion of similar classes. Experimental results on two commonly-used datasets show that our model consistently outperforms other competitive baselines.