  Discrete versions of the Painleve equations (dPII and qPII) over finite
fields are studied. We first show that they are well defined by extending the
domain according to the theory of the space of initial conditions, taking the
dPII equation as an example. Then we define them over the field of p-adic
numbers and see that they have a property that is similar to the good reduction
of dynamical systems modulo a prime. This property is called 'almost good
reduction'. We study the q-discrete analogue of the Painleve II equation in
this paper, following the method in our previous work (arXiv: 1206.4456), in
which the discrete Painleve II equation has been treated. We can consider
almost good reduction as an arithmetic analogue of the singularity confinement
test. We can also obtain special solutions over finite fields from those
defined over fields of characteristic zero.
  (v2: excluded the review of arXiv: 1206.4456) (v3: several typos are
corrected) (v4: final version to appear in J. Nonlin. Math. Phys.)
