An example of a non-paracompact topological space
$\Bbb N^I$ for $I$ uncountable is not normal ($T_4$) as shown by Stone way back. So a fortiori not paracompact as a paracompact Hausdorff space is normal. A reference to this fact online can be found on Dan Ma's nice blog. For the same reason the Sorgenfrey plane is not paracompact, despite being a square of a paracompact space etc.
The long line (and also the simpler $\omega_1$ in the order topology) is more "interesting" in that it is hereditarily normal and countably paracompact but not paracompact. So it's "sharper", as it were, closer to paracompact.