The answer seems to be yes.

Any maximal independent(=no two its vertices share an edge) set is a minimal dominating set. It exists by Zorn’s lemma.

The condition on finite degrees is not used…

======================

**This is a previous (wrong) answer, as I misread the definition of a dominating set.**

Choose $V=\{1,2,\dots\}$ and let the edges be $e_i=\{i,i+1,\dots\}$. The dominating sets are precisely infinite ones.