When I open the passenger door by slightly pulling the exterior handle everything works fine, meaning that the window goes down and stays down until I close the door.
The problem only occurs if someone pulls the exterior handle all the way to the end of its travel.
In that scenario the window goes down but goes up again as soon as you leave the door handle to get inside the car and therefore the window hit the door surround...

 

There are 4 micro-switches that control the window drop and raise functions. Once the window drops from the exterior door handle microswitch (which sounds like it's working), the one in the latch mechanism keeps it down until the door is shut and then that switch closing let's it go back up. Sounds like it's that one that has gone bad.

 

When you open the door, does the window rise again?

The same thing happened to me in one of the 997 I had, it is the microswitch of the door lock. Although you can change the microswitch only, the lock is cheap, since it is the same part number as the lock on a Golf V for example, it is directly a Volkswagen part number.

 

How old is your battery? the windows have been known to get a little buggy with old batteries. Sometimes they need to be reprogrammed. I think there's some procedure in the manual.