Is this situation legal:

Player A controls 4 characters, and one of them is Beric Dondarrion (IG), and plays with The Brotherhood Without Banners Agenda.

Player B reveals Wildfire Assault plot.

Can player A choose 3 characters other than Beric Dondarrion, and to have all 4 characters remain in play?

I will appreciate explanation.

Thanks in advance.

Yes. This interaction works.

When a character "cannot be killed," it means 2 separate things: 1) That character cannot be chosen to die when choosing targets for an effect, and 2) That character doesn't die when a general or un-targeted kill effect resolves.

Now, the thing to recognize here is that #1 has nothing to do with Wildfire and CBK. #1 only stops an effect saying "choose and kill" (or some permutation of that) from choosing the CBK character as a target. Wildfire doesn't do that. Wildfire chooses 3 characters, then kills the rest. Not being allowed to choose a CBK character to die is not the same as being forced to choose a CBK to live. So there is no interaction that makes a player choose Beric as one of his 3 characters for Wildfire.

At that point, #2 kicks in. The "kill all characters not chosen" part of Wildfire is a general, un-targeted kill effect. Since Beric cannot be killed, he simply doesn't die when it resolves. So by choosing the other 3 characters, the player with Beric can end up with all 4 characters in play after Wildfire resolves.

Think of it this way: Say that I have Beric, the BwB agenda, and an Iron Mines on the table. Valar is revealed. Just because Beric cannot be killed, I am not forced to do anything special to make sure he lives - like save him. That's fairly clear in the Beric/Valar/Iron Mines situation, right? The Beric/Wildfire situation isn't really any different. I don't have to do anything special to make sure he lives - like choose him as one of my 3 that won't be killed.