  Inspired by a recent novel work of Good and Meddaugh, we establish
fundamental connections between shadowing, finite order shifts, and ultrametric
complete spaces. We develop a theory of shifts of finite type for infinite
alphabets. We call them shifts of finite order. We develop the basic theory of
the shadowing property in general metric spaces, exhibiting similarities and
differences with the theory in compact spaces. We connect these two theories in
the setting of zero-dimensional complete spaces, showing that a uniformly
continuous map of an ultrametric complete space has the finite shadowing
property if, and only if, it is an inverse limit of a system of shifts of
finite order satisfying the Mittag-Leffler Condition. Furthermore, in this
context, we show that the shadowing property is equivalent to the finite
shadowing property and the fulfillment of the Mittag-Leffler Condition in the
inverse limit description of the system. As corollaries, we obtain that a
variety of maps in ultrametric spaces have the shadowing property, such as
similarities and, more generally, maps which themselves, or their inverses,
have Lipschitz constant 1. Finally, we apply our results to the dynamics of
$p$-adic integers and $p$-adic rationals.
