This space is more commonly called the Michael line $\mathbb{M}$, see this blog post for an online introduction. It has proofs that $\mathbb{M}$ is paracompact but not Lindelöf nor metrisable.
To see it is paracompact: take an open cover $\mathcal{U}$ of $\mathbb{M}$ and we can assume WLOG that it consists of basic open subsets, so either Euclidean open, or singletons from the irrationals. Call the first subcollection of Euclidean open covering sets $\mathcal{U}’$ with union $Y$, then $\mathcal{U}’$ is a standard open cover of $Y$ (which is paracompact, being metrisable) so $\mathcal{U}’$ has locally finite (in the Euclidean sense) refinement $\mathcal{U}’’$. Then this collection together with all $\\{x\\}$, $x \
otin Y$ is a locally finite open refinement of $\mathcal{U}$. So $\mathbb{M}$ is paracompact.