Question: I'm looking for some examples which could show that normal topological space doesn't imply the space is paracompact.

Thanks in advance.

Answer: 
The space $\omega\_1$, the set of countable ordinals, in the order topology, is hereditarily normal (as all ordered spaces), countably (para)compact, collectionwise normal, but not paracompact, because a paracompact countably compact is compact, which $\omega\_1$ is not. 

The nice resource [$\pi$-base](https://topology.jdabbs.com/spaces?q=T_4%20%2B%20~Paracompact) has some more examples, some derived from this one (like the long line), others like Dowker spaces (a normal not countably paracompact space, which are harder to come by).

Bing's spaces like "G" (nicely explained [here](https://dantopology.wordpress.com/2016/12/05/a-stroll-in-bings-example-g/) and "H" (see [here](https://dantopology.wordpress.com/2014/02/02/bings-example-h/)) are also classics, introduced to explore the gap between normality-like properties and paracompact-like properties. I think $\omega\_1$ is the easiest example though.
An example is built here, have you read it:
<https://projecteuclid.org/download/pdf_1/euclid.bams/1183549052>