Next to the town of Stratford, lies the Forest of Arden that Shakespeare drew upon in writing about the forest in As You Like It. In Shakespeare's time, the Forest of Arden was rich in trees and richly populated with cottages, including cottages of "minor gentry" of the sort that Rosalind and Celia bought for their own abode. On Shakespeare's annual returns to Stratford-Upon-Avon, he undoubtedly reacquainted himself with the pleasures of his own Arden Forest.
Individuals in the villages leased land from the owner, one Robert Arden, living in Wilmcote. Shakespeare's own grandfather leased more than 140 acres from Robert Arden in the Forest of Arden. It was Shakespeare's father, John, who first left the forest and moved to the town of Stratford to learn the trade of glover.
Hath not old custom made this life more sweet
Than that of painted pomp? Are not these woods
More free from peril than the envious court?
Here feel we but the penalty of Adam,
The seasons' difference, as the icy fang
And churlish chiding of the winter's wind,
Which, when it bites and blows upon my body,
Even till I shrink with cold, I smile and say
'This is no flattery: these are counsellors
That feelingly persuade me what I am.' (II.i)
Yet there is contradictory scholarly opinion that the Forest of Arden is actually the forest of the Ardennes, which is situated in Belgium, part of the Low Countries. What evidence there is for thinking Shakespeare went so far afield for the model of Arden Forest is not made clear.
Yes, there was an actual forest called "Forest of Arden" in the book. It was situated near the Arden's home and this forest was especially important to the whole family for many generations and generations.
Shakespere compared the forest to the bibilical Eden, which is found in the first chapter of the bible, Genesis. Life in it was compared to the "fallen world" outside and there were similar allusions to the story in the Bible. This forest was a exotic and a magical place to behold, where strange apparitions and coincidences appear out of nowhere, where people are transformed spritually and when love is all around, what a lovely and enchanting environment to indulge!
Why yes, there is an actual forest called Arden that was very important to Shakespeare and his family. His mother's people, the Ardens, owned it. Her family has history dating well back in time to do with the land. There was a female ancestor who was an abbess at the local nunnery. However, the family was Catholic while Queen Elizabeth made being a Catholic a crime. Jesuit missionaries, some of them cousins of Shakespeare, hid out in the forest of Arden to lose the Queen's officers who were trying to hunt them down. Some of Shakespeare's family, who were Ardens were tortured and killed because of their religious affiliation.
And so, the answer is also no. It would have been politically indiscreet for Shakespeare to create a world where the good and the decent were in the forest of Arden while the crimes and foul-play were happening at court. And therefore the forest of Ardenne is to be found fictionally in France.
For more information see Michael Wood's In Search of Shakespeare.