  We prove thatthe Banach space $(\oplus_{n=1}^\infty \ell_p^n)_{\ell_q}$,
which is isomorphic to certain Besov spaces, has a greedy basis whenever $1\leq
p \leq\infty$ and $1<q<\infty$. Furthermore, the Banach spaces
$(\oplus_{n=1}^\infty \ell_p^n)_{\ell_1}$, with $1<p\le \infty$, and
$(\oplus_{n=1}^\infty \ell_p^n)_{c_0}$, with $1\le p<\infty$ do not have a
greedy bases. We prove as well that the space $(\oplus_{n=1}^\infty
\ell_p^n)_{\ell_q}$ has a 1-greedy basis if and only if $1\leq p=q\le \infty$.
