Does $G$ with $\delta(G)\geq \aleph_0$ contain $k$-regular sub-edge-sets?

Let $$G=(V,E)$$ be an infinite, simple, undirected graph, such that for all $$v\in V$$ we have $$\text{deg}(v) \geq \aleph_0$$. Given an integer $$k\geq 1$$, is there always $$E^{(k)}\subseteq E$$ such that $$(V, E^{(k)})$$ is $$k$$-regular, that is, every vertex has exactly $$k$$ neighbors?

The complete bipartite graph $$K_{\aleph_0,\aleph_1}$$ does not contain a $$k$$-regular spanning subgraph for any positive integer $$k$$.