  We consider nonlocal minimal surfaces in a cylinder with prescribed datum
given by the complement of a slab. We show that when the width of the slab is
large the minimizers are disconnected and when the width of the slab is small
the minimizers are connected. This feature is in agreement with the classical
case of the minimal surfaces.
  Nevertheless, we show that when the width of the slab is large the minimizers
are not flat discs, as it happens in the classical setting, and, in particular,
in dimension $2$ we provide a quantitative bound on the stickiness property
exhibited by the minimizers.
  Moreover, differently from the classical case, we show that when the width of
the slab is small then the minimizers completely adhere to the side of the
cylinder, thus providing a further example of stickiness phenomenon.
