Download GEDCOM for Individual.

You can only export from a profile you have added to Geni. The export you can do is limited to a max number of profiles, so it is possible but you must choose the export profile with some care so that that branch of the tree will be included. If you are able to add a close relative to the profile you interested in you could export from that profile.
You should first try to solve as many tree and data conflicts for the branches you want to see in the export (tree and data conflicts will trigger errors or most importing programs)