Dimension quotient space of isomorphic Banach spaces

Question

Let $X,Y$ be two infinite dimensional isomorphic Banach spaces such that $Y$ is a proper subspace of $X$. Is it true that the quotient space $X/Y$ has finite dimension?

Ps. Here, "isomorphic" means "linearly homeomorphic".

Answer 1

Not even remotely true. More than, there's a

Theorem (Pelczynski, 1950s). Every closed subspace in $\ell^p$ with closed complement is isomorphic to $\ell^p$, $1 \leq p < \infty$.

Answer 2

I don't think this is correct. My Counter example : The Calkin algebra. (There is a Wikipedia article about it)

I now realize that only the German Wikipedia article mentions that the Calkin algebra has $2^\aleph_0$ pairwise orthogonal projections, which should imply that it's of infinite dimension. (Taking for granted you use a separable Hilbert space)