  Let $X$ be a ball Banach function space on ${\mathbb R}^n$. Let $\Omega$ be a
Lipschitz function on the unit sphere of ${\mathbb R}^n$,which is homogeneous
of degree zero and has mean value zero, and let $T_\Omega$ be the convolutional
singular integral operator with kernel $\Omega(\cdot)/|\cdot|^n$. In this
article, under the assumption that the Hardy--Littlewood maximal operator
$\mathcal{M}$ is bounded on both $X$ and its associated space, the authors
prove that the commutator $[b,T_\Omega]$ is compact on $X$ if and only if
$b\in{\rm CMO}({\mathbb R}^n)$. To achieve this, the authors mainly employ
three key tools: some elaborate estimates, given in this article, on the norm
in $X$ of the commutators and the characteristic functions of some measurable
subset,which are implied by the assumed boundedness of ${\mathcal M}$ on $X$
and its associated space as well as the geometry of $\mathbb R^n$; the complete
John--Nirenberg inequality in $X$ obtained by Y. Sawano et al.; the generalized
Fr\'{e}chet--Kolmogorov theorem on $X$ also established in this article. All
these results have a wide range of applications. Particularly, even when
$X:=L^{p(\cdot)}({\mathbb R}^n)$ (the variable Lebesgue space),
$X:=L^{\vec{p}}({\mathbb R}^n)$ (the mixed-norm Lebesgue space),
$X:=L^\Phi({\mathbb R}^n)$ (the Orlicz space), and $X:=(E_\Phi^q)_t({\mathbb
R}^n)$ (the Orlicz-slice space or the generalized amalgam space), all these
results are new.
