Generalizing a theorem of Ph. Dwinger, we describe the ordered set of all (up
to equivalence) zero-dimensional locally compact Hausdorff extensions of a
zero-dimensional Hausdorff space. Using this description, we find the necessary
and sufficient conditions which has to satisfy a map between two
zero-dimensional Hausdorff spaces in order to have some kind of extension over
two given Hausdorff zero-dimensional local compactifications of these spaces;
we regard the following kinds of extensions: continuous, open, quasi-open,
skeletal, perfect, injective, surjective. In this way we generalize some
classical results of B. Banaschewski about the maximal zero-dimensional
compactification. Extending a recent theorem of G. Bezhanishvili, we describe
the local proximities corresponding to the zero-dimensional Hausdorff local
compactifications.