My example of a locally compact space which is not normal ,is the Katetov space. this space is defined as follows:

$K$ =$\beta\mathbb{R}$-($cl_{\beta \mathbb{R}}$$\mathbb{N}$-$\mathbb{N}$). this space has The countable subset $\mathbb{N}$ as a closed subset with any accumulation point in $K$. that's why, this space is not countanly compact. but this space is pseudocompact.(You can easily check this claim). then this space is not normal. because a normal pseudocompact space is countably compact.

My example of a locally compact space which is not normal ,is the Katetov space. this space is defined as follows:

$K$ =$\beta\mathbb{R}$-($cl_{\beta \mathbb{R}}$$\mathbb{N}$-$\mathbb{N}$). this space has The countable subset $\mathbb{N}$ as a closed subset with any accumulation point in $K$. that's why, this space is not countably compact. but this space is pseudocompact.(You can easily check this claim). then this space is not normal. because a normal pseudocompact space is countably compact.