If a space is locally compact Hausdorff then it is completely regular and being second-countable implies being metrizable. At the same time a locally compact metric space does not have to be second-countable (at least I do not remember we proved anything like that). For a metric space being separable, Lindelöf or second-countable is equivalent. So, we need an example of a locally compact metric space which is neither one (it can still be first-countable). I know couple spaces that are first-countable but not separable, Lindelöf or second-countable. Both are Hausdorff and locally compact. The first is $S_\Omega$ , not metrizable (§28, page 181: it is limit point compact but not compact). The second is a discrete uncountable space, metrizable.