  We show that if $1<p\neq 2<\infty$, then any isometry of the
$p$-convexification of the combinatorial Banach space associated with a
hereditary family of finite subsets of $\mathbb{N}$ containing the singletons
is given by a signed permutation of the canonical basis. In the case of a
generalized Schreier family, the result also holds for $p=2$, and every
isometry is diagonal. These results are deduced from more general theorems
concerning combinatorial-like Banach spaces.
