Let $G=(V,E)$ be an infinite graph such that $|V| = \kappa$ for some infinite cardinal $\kappa$, and every $v\in V$ has degree $\kappa$. Does $G$ have a perfect matching?
Well, it seems like the following should work:
Let us well order $V$ such that for every $v\in V$ set of $u$ such that $u < v$ has cardinality less then $|V|$. Now we are using transfinite induction to produce matching:
at each step if we are looking at vertex $v$ than if it is already in some pair with one of the previous vertices then just skip it and, otherwise, match it with smallest (under $<$) vertex which is connected with $v$ and not yet chosen. This vertex obviously exists(we already used less then $|V|$ vertices) and so we got our matching.