  We construct new families of smooth Fano fourfolds with Picard rank $1$ which
contain open $\Bbb A^1$-cylinders, that is, Zariski open subsets of the form $Z
\times \Bbb A^1$, where $Z$ is a quasiprojective variety. In particular, we
show that every Mukai fourfold of genus $8$ is cylindrical and there exists a
family of cylindrical Gushel-Mukai fourfolds.
