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…
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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.