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Yes, Animal Crossing: Pocket Camp is available as a free download.
You can play the game for free, but you can also purchase in-game currency called Leaf Tickets using real money.
Note: You can also acquire Leaf Tickets for free.
For details on Leaf Tickets, see About Leaf Tickets within Common Questions.
Note: A persistent Internet connection is required to play Animal Crossing: Pocket Camp.
*Cellular data charges may apply.