  We show that, if $E$ is a Banach space with a basis satisfying a certain
condition, then the Banach algebra $\ell^\infty({\cal K}(\ell^2 \oplus E))$ is
not amenable; in particular, this is true for $E = \ell^p$ with $p \in
(1,\infty)$. As a consequence, $\ell^\infty({\cal K}(E))$ is not amenable for
any infinite-dimensional ${\cal L}^p$-space. This, in turn, entails the
non-amenability of ${\cal B}(\ell^p(E))$ for any ${\cal L}^p$-space $E$, so
that, in particular, ${\cal B}(\ell^p)$ and ${\cal B}(L^p[0,1])$ are not
amenable.
