Example of a separable, locally-compact metric space which is not $\sigma$-compact

I am looking for an example of a separable, locally-compact metric space which is not $\sigma$-compact.

At first I thought I could show that if a metric space is separable and locally-compact, then it must be $\sigma$-compact. But I could not show it and I haven't found any theorem that implies that. So I figured there must be an example of a space which is metrizable, separable, locally-compact but not $\sigma$-compact. Clearly such a space cannot be compact, so I am looking for a locally-compact non-compact metric space.

Can someone give me such an example?

Note that every second-countable locally compact Hausdorff space $X$ is σ-compact.
• Proof. The family $\mathcal B$ of all open $U \subseteq X$ with compact closure forms a base for $X$ by local compactness. By second-countability some countable subfamily $\mathcal B_0$ of $\mathcal B$ is itself a base for $X$. But then $X = \bigcup \{ \overline U : U \in \mathcal B_0 \}$, a countable union of compact sets, so $X$ is σ-compact.
• @AvivEshed I am not using the countable dense set to construct the base, but the knowledge that having a countable dense set means that the metric space is second-countable. It is a fact (though unfortunately not commonly taught) that given any base $\mathcal B$ for a second-countable topological space you can find a countable subfamily $\mathcal B_0$ of $\mathcal B$ which is also a base for $X$. The basic idea for this can be seen in this answer of mine. – Meta-мета-μετα-meta-мета-μετα Aug 23 '15 at 11:33