As we rely more and more on machine learning models for real-life decision-making, being able to understand and trust the predictions becomes ever more important. Local explainer models have recently been introduced to explain the predictions of complex machine learning models at the instance level. In this paper, we propose Local Rule-based Model Interpretability with k-optimal Associations (LoRMIkA), a novel model-agnostic approach that obtains k-optimal association rules from a neighbourhood of the instance to be explained. Compared with other rule-based approaches in the literature, we argue that the most predictive rules are not necessarily the rules that provide the best explanations. Consequently, the LoRMIkA framework provides a flexible way to obtain predictive and interesting rules. It uses an efficient search algorithm guaranteed to find the k-optimal rules with respect to objectives such as confidence, lift, leverage, coverage, and support. It also provides multiple rules which explain the decision and counterfactual rules, which give indications for potential changes to obtain different outputs for given instances. We compare our approach to other state-of-the-art approaches in local model interpretability on three different datasets and achieve competitive results in terms of local accuracy and interpretability.