  We show that for every positive integer $n$ there exists a simple group that
is of type $\mathrm{F}_{n-1}$ but not of type $\mathrm{F}_n$. For $n\ge 3$
these groups are the first known examples of this kind. They also provide
infinitely many quasi-isometry classes of finitely presented simple groups. The
only previously known infinite family of such classes, due to Caprace--R\'emy,
consists of non-affine Kac--Moody groups over finite fields. Our examples arise
from R\"over--Nekrashevych groups, and contain free abelian groups of infinite
rank.
