Nah you can't permanently lose a buddy on a mission, the only thing that happens is you'll receive a notice saying that they're near death and Ocelot or Miller will automatically fulton them out...they will be in the Sick Bay until they heal...the length of time before they're released is contingent on your Medical Staff's level.Other dude made it seem like she died on deployment though on a regular mission.
I didn't think you could lose a buddy permanently just by using them on a mission.
Maybe I'm wrong but that'd be kinda wack cause in order to do any side ops with them or use them in the future you'd have to delete your entire save.
I hope there's no permanent fail state like that. I was hopin to be able to use her still after I beat the story regardless of what happens.
On deployment though? I didn't know you could deploy Quiet on a mission with other staff I thought she was in the Brig when she isn't with you.
I haven't seen or tested it myself, but I think the only way to permanently kill her is through doing so during the story mission.