  Miculescu and Mihail in 2008 introduced the concept of a \emph{generalized
iterated function system} (GIFS in~short), a particular extension of the
classical IFS. The idea is that, instead of families of selfmaps of a metric
space~$X$, GIFSs consist of maps defined on a finite Cartesian {$m$-th power}
$X^m$ with values in $X$ (in such a case we say that a GIFS is \emph{of order}
$m$). It turned out that a great part of the classical {Hutchinson theory} has
natural counterpart in this GIFSs' framework. On the other hand, there are
known only few examples of~fractal sets which are generated by GIFSs, but which
are not IFSs' {attractors}.
  In the paper we study $0$-dimensional compact metrizable spaces from the
perspective of GIFSs' theory. We prove that each such space $X$ (in particular,
countable with limit {scattered} height) is homeomorphic to the~attractor of
some GIFS on the real line. Moreover, we prove that $X$ can be embedded into
the real line $\R$ as {the attractor of some} GIFS of order $m$ and (in the
same time) {a nonattractor} of any GIFS of order $m-1$, as well as it can be
embedded as {a nonattractor of any GIFS}. {Then} we show that a relatively
simple modifications of $X$ deliver spaces whose each connected component is
"big" and which are GIFS's { attractors} not homeomorphic with IFS's
{attractors}. Finally, we use obtained results to show that a generic compact
subset of a Hilbert space is not {the} attractor of any Banach GIFS.
