There's no correlation between Face ID and the internet (note that it works in Airplane mode too) so, it's not an internet issue at all - that's for sure. Face ID is purely a way to gain access to your phone using your face as authentication - your unique face is your password. Seeing that it uses exclusively infrared light which eyes cannot see typically, I wonder if you're one of the rare ones that can sense it a little. And in complete darkness your sensitivity would go up from the lack of distraction from other light sources...
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Well, isn't she trying to find out if someone else has experienced this - she has to ask to find out, doesn't she...?