In this paper, we investigate the following Keller–Segel system with flux limitation [Formula: see text] under no-flux boundary conditions in a ball [Formula: see text]), where [Formula: see text] is positive and [Formula: see text] with [Formula: see text]. It is proved that the problem (⋆) possesses a unique classical solution that can be extended in time up to a maximal [Formula: see text]. Moreover, it is shown that the above solution is global and bounded when either [Formula: see text] if [Formula: see text] and [Formula: see text], or [Formula: see text] if [Formula: see text] and [Formula: see text]. We point out that when [Formula: see text], our result is consistent with that of [N. Bellomo and M. Winkler, Comm. Partial Differential Equations 42 (2017) 436–473].