  This paper aims to study isometries of the $1$-Wasserstein space
$\mathcal{W}_1(\mathbf{G})$ over Carnot groups endowed with horizontally
strictly convex norms. Well-known examples of horizontally strictly convex
norms on Carnot groups are the Heisenberg group $\mathbb{H}^n$ endowed with the
Heisenberg-Kor\'anyi norm, or with the Naor-Lee norm; and $H$-type Iwasawa
groups endowed with a Kor\'anyi-type norm. We prove that on a general Carnot
group there always exists a horizontally strictly convex norm. The main result
of the paper says that if $(\mathbf{G},N_{\mathbf{G}})$ is a Carnot group where
$N_{\mathbf{G}}$ is a horizontally strictly convex norm on $\mathbf{G}$, then
the Wasserstein space $\mathcal{W}_1(\mathbf{G})$ is isometrically rigid. That
is, for every isometry
$\Phi:\mathcal{W}_1(\mathbf{G})\to\mathcal{W}_1(\mathbf{G})$ there exists an
isometry $\psi:\mathbf{G}\to \mathbf{G}$ such that $\Phi=\psi_{\#}$.
