Not totally sure but the width of bullet is the same so I dont see why not. I'm lazy so I have a set of dies for both so I dont have to re-adjust everytime I change calibers.
EDIT: Just did a quick search and the consensus is NO. The 9mm case is tapered and the .380 is not.
Many people seem to believe that 380 to 9mm is like 38 to 357, but it's not. It's not just a shorter version, and you can't use the same dies. Doesn't matter whether you are using regular dies in a loading press or a Lee Loader.