  We prove that the class of reflexive asymptotic-$c_0$ Banach spaces is
coarsely rigid, meaning that if a Banach space $X$ coarsely embeds into a
reflexive asymptotic-$c_0$ space $Y$, then $X$ is also reflexive and
asymptotic-$c_0$. In order to achieve this result we provide a purely metric
characterization of this class of Banach spaces. This metric characterization
takes the form of a concentration inequality for Lipschitz maps on the Hamming
graphs, which is rigid under coarse embeddings. Using an example of a
quasi-reflexive asymptotic-$c_0$ space, we show that this concentration
inequality is not equivalent to the non equi-coarse embeddability of the
Hamming graphs.
