In France, we also have several games from America.
The most common is by far the racoon, and it is incredible how fast this animal is spreading across the country ! They were imported by American soldiers as mascots and most of them were released in the wild at the end.
Unfortunately, we can't hunt them like in America. First of all, electronic callers are prohibited for hunting and even though we have a large number of hounds native from the country, it's forbidden to hunt during the night. Sometimes in driven hunt, dogs catch one but it's nothing in comparison to what we see on cameras.
Trapping is possible but very few people practice it.
We shoot them when the opportunity arise, but it's not common during the legal times.
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The musk rat was also quite common, but it seems they are gone now, at least where I am. From what I have heard, they don't get along with nutrias. The last species is also more and more common where I stay, but not out of control like raccoons. Both were imported for their fur and escaped from farms where they were raised.
In some place, the American mink is also present, but I have never seen one.