[2019] You should be able to open menus when you have been downed.

Joined a stronghold last night, halfway through I got downed in a no respawn area and so did the rest of my team except one guy.

That one guy went afk or something. I was stuck staring at the critical damage screen for a good 15 minutes before I manually exited Anthem from my PS4's home screen, reloaded it and hit new session rather than resume, then loaded and exited freeplay to get the items I'd picked up before that.