You don't mention your age, are you a teenager? Also, maybe, clarify what kind of play you are talking about? -- Tickle? Hug? Joke around? Play a game? (Or something more overtly sensual?) It's pretty natural for someone to get erect when he's with his girlfriend in certain situations.
Just be thankful your penis works as designed and enjoy it. Your girlfriend should be flattered. :) There may come a day many years from now where you'll wish you could get erect. (I hope not, however!)
I think you've gotten some good thoughts. Erections are normal. If you really don't want one and ya, it could embarrass both of you, I get that, then don't play in those ways. (whatever ways they are). Or wear loose pants so it's not noticeable and step away for a second.