The Arens space is obtained by taking a discrete sequence $\{x_n:n<\omega\}$ converging to $\infty$, then attaching another discrete sequence $\{x_{n,m}:m<\omega\}$ converging to each $x_n$.
This space is $T_1$ as points are closed. What's the strongest $T_n$ separation axiom we can apply?
The space is zero-dimensional as basic open sets are clopen. It's also immediate that the space is countable.
As noted in the preprint at https://arxiv.org/pdf/1306.6086.pdf, all Lindelof zero-dimensional spaces are ultraparacompact, that is, every open cover has an open refinement partitioning the space. Given an open cover, refine it with countably-many clopen sets $\{C_n:n<\omega\}$, then refine it further into the partition $\{C_n\setminus\bigcup_{m<n}C_m:n<\omega\}$.
Ultraparacompact spaces are strongly paracompact (every open cover has a star-finite refinement) and thus paracompact (every open cover has a locally finite refinement).
The space is $T_0$ and regular, giving $T_3$. The space is then $T_2$ and paracompact, giving (fully) $T_4$.
As Patrick points out in the comments, we can also get $T_6$. Countable is hereditary, and thus the space is hereditarily Lindelof. Along with regularity we have perfect normality.