Growth $\beta X\setminus X$ of a Banach space $X$

Concerning the second part of your question, when $X$ is a Banach space, one can always find a "larger" Banach space containing $X$ as a proper, closed subspace.

Indeed, if $Y$ is another Banach space one can endow the direct product $X \times Y$ with a norm making it a Banach space with $X \simeq X \times \{ 0 \}, Y \simeq \{ 0 \} \times Y$ as closed subspaces. Such a norm is certainly not unique, but one example is given by $|| (x, y) || = || x || _{X} + || y || _{Y}$. There exists Banach spaces $Y$ of arbitrary cardinality, since $L^{1}(\Omega)$ is a Banach space of cardinality at least $| \Omega |$ for any set $\Omega$, so one can make $X \times Y$ arbitrarily large too.