Hi Caroline,
although I am not German I think students in Germany apply for a PhD and then get a salary, or they are self funded.
There is a basic system called "Bafög", but I think this is for regular students only, I do not think it would be provided for a PhD. There is also a system called "Stipendium" (scholarship) , which a student can apply for. I am not an expert in this field but I get the impression that this is only very a basic amount of money. These "Stipendia" are provided by various organisations, profit and non-profit, and it may be an idea to look for them, perhaps by doing a google search. Good luck:-)