The .part file is a temporary file that should be deleted or renamed once the download is completed. For some reason, it appears that the download is not being completed.

Thanks! The problem is that it happens always and immediately after starting download, so there is nothing to rename.

On Windows I usually get two files: a placeholder for the real name and then the .part file which grows during the download and then disappears when it is complete.

Maybe try changing the location where you save downloads, it's on the General tabs of the Preferences dialog. If you normally use "Always ask me where to save," you have to change that temporarily in order to access the directory setting. Perhaps that will sort out Firefox's internal settings.