  We introduce "braided" versions of self-similar groups and
R\"over--Nekrashevych groups, and study their finiteness properties. This
generalizes work of Aroca and Cumplido, and the first author and Wu, who
considered the case when the self-similar groups are what we call
"self-identical". In particular we use a braided version of the Grigorchuk
group to construct a new group called the braided R\"over group, which we prove
is of type $F_\infty$. Our techniques involve using so called $d$-ary cloning
systems to construct the groups, and analyzing certain complexes of embedded
disks in a surface to understand their finiteness properties.
