  Building on previous work by Caicedo and the second author, we develop a
method that decides the existence of units of finite order in blocks of
$\mathbb{Z}_p G$ of defect 1. This allows us to prove that if $p$ is a prime
and $G$ is a finite group whose Sylow $p$-subgroup has order $p$, then any unit
$u\in \mathbb{Z} G$ of order $p$ is conjugate to an element of $\pm G$. This is
a special case of the Zassenhaus conjecture. We also prove some new results on
units of finite order in $\mathbb{Z} \mathrm{PSL}(2,q)$ for certain $q$, and
construct a unit of order $15$ in $V(\mathbb{Z}_{(3,5)}\mathrm{PSL}(2,16))$
which is a $3$- and $5$-local counterexample to the Zassenhaus conjecture,
raising the hope that our methods may lead to a global counterexample amongst
simple groups.
