I found the answer on the Debian Forum.
It boils down to this: edit the file /etc/gdm3/greeter.gconf-defaults, uncomment the line # Use a specific background, and change the file name to the name of the file you want to use. (Drop an
image in /usr/share/images/desktop-base.)
Update: other image formats work too, for example .png.
Update 2: If the image file is not displayed (resulting in the GDM screen being a rather unpleasant shade of green), check the file permissions: root must have access (read only).