A Banach space $X$ is called self-saturated if every infinite dimensional subspace of it contains a further subspace isomorphic to $X$. Some classical examples of such spaces are $c_0$ and $\ell_p$-spaces for $1\leq p<\infty$ (Pelczynski, 1960).
Are there any other known examples of self-saturated spaces?
A self-saturated Banach space is called minimal. There is now whole industry for such spaces, however using $\ell_p$ and $c_0$ you may produce some trivial examples by passing to subspaces of these spaces. Thus, for example $c_0$-sums of finite-dimensional Banach spaces are minimal as they always embed into $c_0$.
The most prominent, non-classical example is Schlumprecht's space.