Variational Bayesian (VB) learning has been successfully applied to instantaneous blind source separation. However, the traditional VB learning is restricted to the separation of independent source signals. Moreover, it has the difficulty to recover source signals with a sizable number of samples because of its rapidly increasing computational requirement. To overcome such shortcomings, frame-based VB (FVB) learning is proposed to address both independent and dependent source separation with a large number of samples in this paper. Specifically, a Gaussian process (GP) is employed to model independent or dependent source signals. To our knowledge, GP has been only used to model each of independent source signals. For dependent source signals, this paper proposes a novel modeling process: initial source signals are zigzag concatenated into a long serial and GP is then used to model it. In order to obtain a reliable covariance function for GP, first, we apply singular value decomposition to give initial estimated source signals and then we select an appropriate covariance function with which GP can perfectly fit them. In order to alleviate the computational burden of VB learning, we split observed signals into frames, and then model and infer source signals for each frame. Compared with the state-of-the-art algorithms, the experimental results show that the FVB learning has potential to provide improvement in separation performance not only for independent source signals but also for dependent ones, especially for long data records.