The most famous example of a non-paracompact space is the long line. I saw in several places a reference to another example: the Cartesian product of uncountably many copies of an infinite discrete space is not paracompact. Although I have searched for a proof of this fact, I was unable to find one (or to prove it myself). Even the Wikipedia article on paracompact spaces mentions this fact without providing a reference.
Can someone explain why such a space is not-paracompact?
$\Bbb N^I$ 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.