  We show that $\Ext^2(\ell_2, \ell_2)\neq 0$ in the category of Banach spaces.
This solves a sharpened version of Palamodov's problem and provides a solution
to the second order version of Palais problem. We also show that
$\Ext^2(\ell_1, \K)\neq 0$ in the category of quasi Banach spaces which solves
the four-space problem for local convexity.
