Fix some $n \geq 1$ and some prime $p$. I'm looking for finite $p$-groups $G$ and finite-dimensional complex representations $V$ of $G$ with the following two properties:

The abelianization of $G$ has rank $n$; for instance, it could be $(\mathbb{Z}/p)^n$.

For all nonidentity $g \in G$ and all nonzero $v \in V$, we have $g(v) \neq v$. In other words, $1$ is not an eigenvalue for the action of $g$ on $V$.

Here are the cases I know how to deal with:

For $n=1$ and an arbitrary prime $p$, you can take the cyclic group of order $p$ and the $1$-dimensional representation where the generator acts as rotation by $2\pi/p$.

For $n=2$ and $p=2$, you can let $G$ be the finite quaternion group and $V$ be the unique $2$-dimensional irreducible representation of $G$.