French Guyana is not part of the Schengen Area. So does a third country national living with a German residency permit in Germany need to apply for a visa to go there?
Where to apply for such document?
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According to the French government site, a visa national who has a Schengen residence permit or long-stay visa is exempt from the visa requirement for a short stay (or up to 90 days) in French Guyana even though French Guyana is not part of the Schengen area:
You need a visa unless you
[Are in] possession of a long-stay visa or hold a valid residence permit issued by a Schengen member state, [or]
Hold a residence permit for Andorra or Monaco.
(There are some other exceptions, but they are beyond the scope of this question.)
French Guyana uses the same visa exemption list as the rest of France even though it is not part of the Schengen area. We see from the above information that someone who has a German residence permit does not need a visa for a short stay French Guyana in any case, but it's also possible that such a person would not need a visa even without the residence permit, because the person might be a national of an "Annex II" country whose citizens are exempt from the short-stay visa requirement.
The text above was generated by assuming that the traveler is a national of Bhutan, an Annex I nationality (that is, a "visa national"). To see what the visa tool would say for someone with an Annex II nationality, we can assume that the person is Canadian, in which case the tool yields this result:
You do not need a visa.
There follows a disclaimer box with the text
Nevertheless, you must be able to show some documents when you are controlled at the border. For more information, please click here
The link takes you to a page describing several documents serving as evidence that you fulfill the conditions of a short-stay visitor, including financial means, accommodation, return ticket or additional financial means to obtain one, professional status (for business visitors), and insurance. It says that these must be presented "at the request of the Border Police upon your arrival in France." In my experience, the border police do not typically ask for these documents, but then I've never traveled to French Guyana, so I have no idea whether the practice there might be different.