Adapting Krishna's answer, which uses AVAssetExportSession to save the MPMediaItem to a file, you can do something like the following in Swift 3:
/// Export MPMediaItem to temporary file.
///
/// - Parameters:
///   - assetURL: The `assetURL` of the `MPMediaItem`.
///   - completionHandler: Closure to be called when the export is done. The parameters are a boolean `success`, the `URL` of the temporary file, and an optional `Error` if there was any problem. The parameters of the closure are:
///
///   - fileURL: The `URL` of the temporary file created for the exported results.
///   - error: The `Error`, if any, of the asynchronous export process.
func export(_ assetURL: URL, completionHandler: @escaping (_ fileURL: URL?, _ error: Error?) -> ()) {
    let asset = AVURLAsset(url: assetURL)
    guard let exporter = AVAssetExportSession(asset: asset, presetName: AVAssetExportPresetAppleM4A) else {
        completionHandler(nil, ExportError.unableToCreateExporter)
        return
    }
    let fileURL = URL(fileURLWithPath: NSTemporaryDirectory())
        .appendingPathComponent(NSUUID().uuidString)
        .appendingPathExtension("m4a")
    exporter.outputURL = fileURL
    exporter.outputFileType = "com.apple.m4a-audio"
    exporter.exportAsynchronously {
        if exporter.status == .completed {
            completionHandler(fileURL, nil)
        } else {
            completionHandler(nil, exporter.error)
        }
    }
}
func exampleUsage(with mediaItem: MPMediaItem) {
    if let assetURL = mediaItem.assetURL {
        export(assetURL) { fileURL, error in
            guard let fileURL = fileURL, error == nil else {
                print("export failed: \(error)")
                return
            }
            // use fileURL of temporary file here
            print("\(fileURL)")
        }
    }
}
enum ExportError: Error {
    case unableToCreateExporter
}
Or, in Swift 2:
/// Export MPMediaItem to temporary file.
///
/// - Parameters:
///   - assetURL: The `assetURL` of the `MPMediaItem`.
///   - completionHandler: Closure to be called when the export is done. The parameters are a boolean `success`, the `URL` of the temporary file, and an optional `Error` if there was any problem. The parameters of the closure are:
///
///   - fileURL: The `URL` of the temporary file created for the exported results.
///   - error: The `Error`, if any, of the asynchronous export process.
func export(assetURL: NSURL, completionHandler: (NSURL?, ErrorType?) -> ()) {
    let asset = AVURLAsset(URL: assetURL)
    guard let exporter = AVAssetExportSession(asset: asset, presetName: AVAssetExportPresetAppleM4A) else {
        completionHandler(nil, ExportError.unableToCreateExporter)
        return
    }
    let fileURL = NSURL(fileURLWithPath: NSTemporaryDirectory())
        .URLByAppendingPathComponent(NSUUID().UUIDString)!
        .URLByAppendingPathExtension("m4a")
    exporter.outputURL = fileURL
    exporter.outputFileType = "com.apple.m4a-audio"
    exporter.exportAsynchronouslyWithCompletionHandler {
        if exporter.status == .Completed {
            completionHandler(fileURL, nil)
        } else {
            completionHandler(nil, exporter.error)
        }
    }
}
func exampleUsage(with mediaItem: MPMediaItem) {
    if let assetURL = mediaItem.assetURL {
        export(assetURL) { fileURL, error in
            guard let fileURL = fileURL where error == nil else {
                print("export failed: \(error)")
                return
            }
            // use fileURL of temporary file here
            print("\(fileURL)")
        }
    }
}
enum ExportError: ErrorType {
    case unableToCreateExporter
}
As you can see, I put it in a temporary folder rather than the Documents folder. Also, I use UUID rather than the number of seconds since some reference date to generate the temporary file. But the idea is basically the same.