  We study the existence of (unmixed) Beauville structures in finite
$p$-groups, where $p$ is a prime. First of all, we extend Catanese's
characterisation of abelian Beauville groups to finite $p$-groups satisfying
certain conditions which are much weaker than commutativity. This result
applies to all known families of $p$-groups with a good behaviour with respect
to powers: regular $p$-groups, powerful $p$-groups and more generally potent
$p$-groups, and (generalised) $p$-central $p$-groups. In particular, our
characterisation holds for all $p$-groups of order at most $p^p$, which allows
us to determine the exact number of Beauville groups of order $p^5$, for $p\ge
5$, and of order $p^6$, for $p\ge 7$. On the other hand, we determine which
quotients of the Nottingham group over $\mathbb{F}_p$ are Beauville groups, for
an odd prime $p$. As a consequence, we give the first explicit infinite family
of Beauville $3$-groups, and we show that there are Beauville $3$-groups of
order $3^n$ for every $n\ge 5$.
