While we don't have a direct way of creating an RSVP cut-off date, what you can do is, on the day of the cut-off, limit the number of guests that can attend the event to however many people are currently attending. This will prohibit any additional guests from RSVPing.
To do this, click on the Manage Invitation button for the event and then click on the Invitation Options button. Towards the bottom of the page will be an option to limit the number of attendees. Turn this ON and then type in the number of guests currently on your guest list.