  In this paper, we study the geometry of non-Archimedean Gromov-Hausdorff
metric. This is the first part of our series work, which we try to establish
some facts about the counterpart of Gromov-Hausdorff metric in the
non-Archimedean spaces. One of the motivation of this work is to find some
implied relations between this geometry and number theory via p-adic analysis,
so that we can use the former as a tool to study the relating arithmetic
aspects.
