Graph convolutional networks (GCN) have attracted increasing attention in hyperspectral images (HSIs) classification because of its excellent capacities in modeling arbitrarily irregular data. The essential aim of GCN-based methods is obtaining a more reliable graph that accurately describes the similarity between graph nodes and makes its representation more discriminative. However, it is a challenging task to get a high-quality graph during the convolution process. In this paper, a novel spectral-spatial dynamic graph convolutional network (SSD-GCN) is proposed for HSIs classification, which not only can adaptively update graph according to the HSI content but also can generate the discriminative node features during the convolution process, by integrating the current spectral-spatial information of nodes and the graph embedding in the previous layers. Unlike the traditional GCN-based methods that directly convert the raw HSI into a graph in the preprocessing process, we further integrate the graph mapping into the network, to reduce the irrelevant information among spectral bands and facilitate node feature learning. In addition, an auxiliary local context-aware feature reconstruction is constructed to enhance the local representational capacities of the node features and alleviate over-smoothing. Extensive experiments compared with state-of-the-art methods on three HSIs datasets, including Pavia University, Salinas, and Kennedy Space Center, demonstrate the effectiveness and superiority of our proposed SSD-GCN method, even with small-sized training data.