  We investigate self-similar solutions of the extended discrete KP hierarchy.
It is shown that corresponding ansatzes lead to purely discrete equations with
dependence on some number of parameters together with equations governing
deformations with respect to these parameters. Some examples are provided. In
particular, it is shown that the well known discrete first Painleve equation
(dPI) and its hierarchy arises as self-similar reduction of Volterra lattice
hierarchy which in turn can be treated as a reduction of the extended discrete
KP hierarchy. It is written down equations which naturally generalize dPI. It
is shown that theses discrete systems describe B\"acklund transformations of
Noumi-Yamada systems of type $A_{2(n-1)^{(1)}}$. We also consider Miura
transformations relating different infinite- and finite-field integrable
mappings. Simplest example of this kind of Miura transformations is given.
