Deterministic lateral displacement (DLD) technology is a newly developed method which can separate microscale and nanoscale particles continuously and efficiently. In this paper, a direct numerical simulation method (i.e. a fictitious domain method) is used to simulate the motion of an elastic particle (modelled as homogeneously elastic body) in the DLD device. The effects of the particle deformability on the critical separation diameter are investigated. Our results indicate that there exists a critical deformability, below which the critical diameter decreases with increasing deformability, whereas beyond which the critical diameter increases with increasing deformability. The reasons are discussed via the consideration of the effects of the particle deformation and the lubrication force on the lateral position of the particle centre point. In addition, our results show that the increase in the gap distance between adjacent posts in both directions or in the longitudinal direction alone leads to the increase in the critical particle size with respect to the gap size, which can be explained by the lateral position of the separation streamline of the undisturbed flow.