In this paper, we develop an asynchronous massive access framework. Besides the multiple-access interference issue, asynchronous transmission induces inter-symbol interference effect of the same user that is determined by the unknown user delay. To solve this problem, we propose an Bayesian receiver where the whole receiver is divided into two modules: the channel-signal decomposition (CSD) module and the delay learning (DL) module. The CSD module demixes the transmit signals of different users by leveraging the bilinear generalized approximate message passing (BiGAMP) algorithm, and the DL module is designed to estimate the time delay of each user based on the Bayesian principle. Additionally, due to the continuity of time delay, the constellation of all possible received user signals consists of lines and curves instead of discrete points, even if the original transmit signals are discrete. To reduce complexity, we introduce a truncation and projection based approximation method to simplify the related message calculation. Numerical results demonstrate the superior performance of the proposed scheme. Particularly, the proposed scheme is able to approach the single-user interference-free bound with known user delay.