  A new hierarchy of Banach spaces $T_k(d,\theta)$, $k$ any positive integer,
is constructed using barriers in high dimensional Ellentuck spaces
\cite{DobrinenJSL15} following the classical framework under which a Tsirelson
type norm is defined from a barrier in the Ellentuck space
\cite{Argyros/TodorcevicBK}. The following structural properties of these
spaces are proved. Each of these spaces contains arbitrarily large copies of
$\ell_\infty^n$, with the bound constant for all $n$. For each fixed pair $d$
and $\theta$, the spaces $T_k(d,\theta)$, $k\ge 1$, are $\ell_p$-saturated,
forming natural extensions of the $\ell_p$ space, where $p$ satisfies
$d\theta=d^{1/p}$. Moreover, they form a strict hierarchy over the $\ell_p$
space: For any $j<k$, the space $T_j(d,\theta)$ embeds isometrically into
$T_k(d,\theta)$ as a subspace which is non-isomorphic to $T_k(d,\theta)$.
