Is there a term for the people who walk around with their phones in their hand?

Nomophobe derived from "nomophobia"

https://en.wikipedia.org/wiki/Nomophobia "Nomophobia is the fear of being out of mobile phone contact".

phubber (urban-dictionary)

The combination of the words, "phone" and "snubber" which means people who lower their heads looking at their smart phones.

https://en.wikipedia.org/wiki/Phubbing

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