The character will fall through the floor on any terrain, in any car, with any variation to 3rd person or 1st person play. It happens 9/10 times and the only way I've found to get out of the falling animation is to log out of the server and then back in again.
I've also tried this with a brand new car to make sure it wasn't something I was putting in it's inventory causing the issue.
This issue is on the livonia map, I haven't tried it on chernarus.