  The purpose of this paper is to lay the foundations for the study of the
problem of when $\Ext^n(X, Y)=0$ in Banach/quasi-Banach spaces. We provide a
number of examples of couples $X,Y$ so that $\Ext^n(X,Y)$ is (or is not ) $0$,
including the first example of a separable Banach space $\mathscr K$ so that
$\Ext^n(\mathscr K, \mathscr K)\neq 0$ for all $n\in \N$. Such space moreover
provides the first example of Banach spaces with infinite homological
dimension/codimension. We also show that the homological dimension/codimension
of Hilbert spaces is infinite. The final section is devoted to compare
$\Ext^2(\cdot, \cdot)$ in Banach and Quasi-Banach spaces.
