iCloud Drive Bird Service always running
I got rid of the problem by removing the folder that contained the file over the terminal. Seems like there was something mixed up.
This is how I did it:

Identify the source of the problem using this command:
brctl log -w --shorten

I looked into it and saw there was one error always repeating and it always tried to upload it.

Because I couldn't find the file via the terminal (because it was never on my disk) I deleted the folder that should have contained the file. It did that using the command
sudo rm -rf name_of_the_folder


I hope that helps someone.