Are Hardcore characters gone forever if they die in PvP?

With the addition of brawling in Patch 1.0.7, the Player vs Player section was added to the Game Guide, which states:

When you die in the Chapel, you’ll reappear there (even if you’re a hardcore character, your death isn’t permanent). You won’t lose any equipment durability for participating (or perishing) in a brawl.

So, no, hardcore characters are not lost!

Blizzard's current position, which will presumably hold until PvP's release, is that due to the rapid death nature of the arena, hardcore characters will not be permanently dead when they die in the PvP arena.

According to the Diablo Wiki, a quote from Wyatt Cheng in an interview:

Wyatt Cheng: Our current position is that you don’t die in PvP in Hardcore mode. So what happened in Diablo II is that anyone who wanted to do Hardcore PvP would basically level a character to level 10, and then they’d find another level 10 character, and they’d fight and one would die. That’s just not really that interesting. So we’re just going to let players engage in PvP.