Deep learning (DL) has achieved breakthrough successes in various tasks, owing to its layer-by-layer information processing and sufficient model complexity. However, DL suffers from the issues of both redundant model complexity and low interpretability, which are mainly because of its oversimplified basic McCulloch–Pitts neuron unit. A widely recognized biologically plausible dendritic neuron model (DNM) has demonstrated its effectiveness in alleviating the aforementioned issues, but it can only solve binary classification tasks, which significantly limits its applicability. In this study, a novel extended network based on the dendritic structure is innovatively proposed, thereby enabling it to solve multiple-class classification problems. Also, for the first time, an efficient error-back-propagation learning algorithm is derived. In the extensive experimental results, the effectiveness and superiority of the proposed method in comparison with other nine state-of-the-art classifiers on ten datasets are demonstrated, including a real-world quality of web service application. The experimental results suggest that the proposed learning algorithm is competent and reliable in terms of classification performance and stability and has a notable advantage in small-scale disequilibrium data. Additionally, aspects of network structure constrained by scale are examined.