  Weakly locally finite division rings were considered in \cite{dbh}, where it
was mentioned that the class of weakly locally finite division rings properly
contains the class of locally finite division rings. In this paper, for any
integer $n\geq 0$ or $n=\infty$, we construct a weakly locally finite division
ring whose Gelfand-Kirillov dimension is $n$. This fact shows in particular
that there exist infinitely many weakly locally finite division rings that are
not locally finite. Further, for the class of weakly locally finite division
rings, we investigate some questions related with the well-known Kurosh Problem
and with one of Herstein's conjectures.
