We show that for each d>0 the d-dimensional Hamming graph H(d,q) has an
orientably regular surface embedding if and only if q is a prime power p^e. If
q>2 there are up to isomorphism \phi(q-1)/e such maps, all constructed as
Cayley maps for a d-dimensional vector space over the field of order q. We show
that for each such pair d, q the corresponding Belyi pairs are conjugate under
the action of the absolute Galois group, and we determine their minimal field
of definition. We also classify the orientably regular embedding of merged
Hamming graphs for q>3