Obviously the UK will have to do something. When they exit, they will have no trade agreements.
They will also have to sort out their issues with worker displacement. Lot of UK folks working in the EU. Lot of EU folks working in the UK. And a lot of EU/UK people working in the UK, but for companies that will need to move operations to an EU member.
But I don't think the EU is just going to say "ok, y'alls can keep the free trade zone with no strings attached"
So either they will get to stay, but then be subject to all of the same trade and product regulations that the #leave campaigners were using as ammunition...or they will have to start setting up their own agreements/joining other partnerships.