No, unfortunately you cannot save an unfinished ticket. You can, however, choose to Continue a current ticket if you do not log out of your account or close the browser. If you exit a new ticket midway through, BYDP will keep a temporary copy of it available until you log out of that session or until you start another new ticket. These tickets can be reopened by clicking the Continue Current button from the Console screen (which will appear next to the New Ticket button):
After submitting the ticket, or if you log out, this button will disappear from the Console screen.