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Multipeer Connectivity does not support operating in the background; it's really that simply. You may be able to get some things to work, but I recommend that you not go down that path because, when you do unsupported things, you run the risk of future changes breaking your code.
The networking stack does, in general, support operating in the background. Where things start to go wrong is if your app gets suspended in the background. Technote 2277 Networking and Multitasking has the details. So, if your already have a way to prevent your app from suspended—for example, it's an audio app that's playing music—you can use other networking APIs in the background quite successfully.
OTOH, if you can't avoid suspending in the background, you will have to look at APIs that let you make progress while you're suspended; that typically means using NSURLSession background session support, possibly in conjunction with UIApplication's background fetch support.
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