We obtain sufficient and necessary conditions for the Choquet-Deny theorem to hold in the class of compactly generated totally disconnected locally compact groups of polynomial growth, and in a larger class of totally disconnected generalized \bar{FC}-groups. The following conditions turn out to be equivalent when G is a metrizable compactly generated totally disconnected locally compact group of polynomial growth:

• The Choquet-Deny theorem holds for G.
• The group of inner automorphisms of G acts distally on G.
• Every inner automorphism of G is distal.
• The contraction subgroup of every inner automorphism of G is trivial.
• G is a SIN group.

We also show that for every probability measure μ on a totally disconnected compactly generated locally compact second countable group of polynomial growth, the Poisson boundary is a homogeneous space of G, and that it is a compact homogeneous space when the support of μ generates G.

The first author was supported by an NSERC Grant.