In the 3rd round of the game Celebrities, one has to communicate a celebrity to his partner by acting/gesturing without words. Is there a name of that activity?
Is it maybe possible to use play mime as a phrase outside the context of theatre?
This game is known as Charades. People can "play Charades" or "play a round of Charades."
If you're not actually playing a game and you're just trying to use gestures to communicate because of a language barrier, you could say something like
He mimed driving a car hoping that the locals would understand he was looking for a taxi.
Conversations were mostly in mime, because he didn't know much Vietnamese.
Here is a "real life" example:
He let us know (mostly in mime since we don’t know French and he didn’t know English) that if we needed anything this was where he lived and he would be happy to help. (Source)
If you say "play mime", as in the title of the article you linked in the question, you're saying something similar to "play at being a mime" or "pretend like you are a mime". We could also say something like "Hey kids, lets play take a nap!"