Vision-language pre-training (VLP) has attracted increasing attention recently. With a large amount of image-text pairs, VLP models trained with contrastive loss have achieved impressive performance in various tasks, especially the zero-shot generalization on downstream datasets. In practical applications, however, massive data are usually collected in a streaming fashion, requiring VLP models to continuously integrate novel knowledge from incoming data and retain learned knowledge. In this work, we focus on learning a VLP model with sequential chunks of image-text pair data. To tackle the catastrophic forgetting issue in this multi-modal continual learning setting, we first introduce pseudo text replay that generates hard negative texts conditioned on the training images in memory, which not only better preserves learned knowledge but also improves the diversity of negative samples in the contrastive loss. Moreover, we propose multi-modal knowledge distillation between images and texts to align the instance-wise prediction between old and new models. We incrementally pre-train our model on both the instance and class incremental splits of the Conceptual Caption dataset, and evaluate the model on zero-shot image classification and image-text retrieval tasks. Our method consistently outperforms the existing baselines with a large margin, which demonstrates its superiority. Notably, we realize an average performance boost of $4.60\%$ on image-classification downstream datasets for the class incremental split.