Hausdorffness assumed.
All the usual suspects don't work: $\mathbb{Z}^\mathbb{R}$, $2^\mathbb{R}$, discrete $\mathbb{R}$, etc.
My reasoning so far: if it is locally compact, then there are separable compact neighborhoods; of these, the ones that I know that are non-metrizable are uncountable copies of compact spaces such as $2$ or $[0,1]$. So, how to modify these so that they are only locally compact? Maybe $\beta\mathbb{Z}$ with a missing point?