  Call a subset of $\mathbf{FIN}_k$ small if it does not contain a copy of
$\langle{P\rangle}$ for some infinite block sequence $P \in
\mathbf{FIN}_k^{[\infty]}$. Gowers' $\mathbf{FIN}_k$ theorem asserts that the
set of small subsets of $\mathbf{FIN}_k$ forms an ideal, so it is sensible to
consider almost disjoint families of $\mathbf{FIN}_k$ with respect to the ideal
of small subsets of $\mathbf{FIN}_k$. We shall show that
$\mathfrak{a}_{\mathbf{FIN}_k}$, the smallest possible cardinality of an
infinite mad family of $\mathbf{FIN}_k$, is uncountable.
