  Let $K$ be a nonempty subset of a Banach space $X$. A mapping $T\colon K\to
K$ is called $\mathfrak{cm}$-nonexpansive if for any sequence
$(u_i)_{i=1}^\infty$ and $y$ in $K$, $\limsup_{i\to\infty}
\sup_{A\subset\{1,\dots, n\}}\|\sum_{k\in A} \big(T u_{i+k} - Ty\big)\|\leq
\limsup_{i\to\infty} \sup_{A\subset\{1,\dots, n\}}\|\sum_{k\in A} (u_{i+k} -
y)\|$ for all $n\in\mathbb{N}$. As a subclass of the class of nonexpansive
maps, its FPP is well-established in a wide variety of spaces. The main result
of this paper is a fixed point result relating
$\mathfrak{cm}$-nonexpansiveness, $\ell_1$ spreading models and Schauder bases
with not-so-large basis constants. As a consequence, we deduce that Banach
spaces with the weak Banach-Saks property have the fixed point property for
$\mathfrak{cm}$-nonexpansive maps.
