We prove that an $m$ out of $n$ bootstrap procedure for Chatterjee's rank correlation is consistent whenever asymptotic normality of Chatterjee's rank correlation can be established. In particular, we prove that $m$ out of $n$ bootstrap works for continuous as well as for discrete data with independent coordinates; furthermore, simulations indicate that it also performs well for discrete data with dependent coordinates, and that it outperforms alternative estimation methods. Consistency of the bootstrap is proved in the Kolmogorov as well as in the Wasserstein distance.