Let $K$ be a compact scattered space and $X=C(K)$ the space of continuous functions on $K$. We want to show that there is no injective bounded linear $T:X\to Y$ into a weakly sequentially complete (w.s.c.) Banach space $Y$ if $K$ has *large*(see below) cardinality.

Let $T:X\to Y$ be as above. $C(K)$ has Pelczynski property (V) and $Y$ is w.s.c., so $T$ is weakly compact. Since $C(K)$ has Dunford-Pettis property, then $T$ is completely continuous. Lastly $C(K)$ contains no copy of $\ell^1$ since $K$ is scattered. Thus, $T$ is compact.

Since $T$ is compact, its range $T(X)$ is separable, so $card(T(X))\leq\mathfrak{c}=card(\mathbb{R})$. $T$ is injective, so $card(X)=card(T(X))$. Together, we have
$$ card(K)\leq card(X)=card(T(X))\leq\mathfrak{c}.$$
However, this leads to a contradiction when $card(K)>\mathfrak{c}$. For example, take $K=[0,\alpha]$ for some ordinal $\alpha$ with the order topology, where the cardinality of $\alpha$ is strictly greater than $\mathfrak{c}$.

Note (2023-02-18): Following the footsteps in this post, we can actually make a slightly more general statement. If $X$ is a Banach space with property (V), $X^*$ has Schur property, and $card(X)>\mathfrak{c}$, then there exists no injective bounded $T:X\to Y$ into a w.s.c. $Y$.

1more comment