It would appear that Lady Gaga is a hermaphrodite apparently, which you can read about here. There is a video of her ‘penis’ making an appearance, which you can see below. It happens at about the 1:11 mark just after she steps off the moped.
I’m calling shenanigans on this, I can’t really explain whatever it is that appears between her legs but if I know Lady GaGa (which I don’t, but still) it’s probably another publicity stunt. Here’s what she supposedly had to say about it, which is why I think this is possibly fake:
“It’s not something that I’m ashamed of, just isn’t something that I go around telling everyone. Yes. I have both male and female genitalia, but I consider myself a female. It’s just a little bit of a penis and really doesn’t interfere much with my life. The reason I haven’t talked about it is that it’s not a big deal to me. Like come on. It’s not like we all go around talking about our vags. I think this is a great opportunity to make other multiple gendered people feel more comfortable with their bodies. I’m sexy, I’m hot. I have both a poon and a peener. Big f*cking deal.”

I’m not familiar with exactly how Lady GaGa speaks, but as an American I doubt she uses the term ‘vag’ or even ‘poon’ and ‘peener,’ it all seems very much faked to me.