The recommendations definitely do have to be in by the deadline. I think they will make exceptions if they come in RIGHT after, and it's only 1 rec, since they know how senior people can be about these things, but I would be terrified if that happened to me because it may result in you being pushed to R2 after all your hard work.
I would sit down with your recommender and impress upon him how important this is. See if there's anything you can do to help him complete them on time - laying out suggested talking points, etc. Ideally the person you chose is someone that gets it and wants to take the time to make sure you succeed here. I know how frustrating it is though... my boss is definitely a "last minute" guy, so no matter what I do, he'll have me biting my nails about them getting in on time.
From the Wharton discussion forum:
All parts of the application need to be submitted before the deadline to guarantee that you make it into round 1. There's a slight chance that if your recommenders submit their applications in between the deadline and the time that your application is checked for completeness, you may still be considered before round 1. But the order in which applications are checked for completeness is random and there is no way to predict whether yours will be checked 1 hour or 1 week after the deadline. So to be safe, I recommend that you communicate the importance of the deadline to your recommenders.
Also, note that you can submit your application before the recommendations are submitted. And you need to do so before the deadline in order to be considered for round 1.