Александр

Библиотека SSZipArchive, нужно, помимо самой библиотеки, импортировать еще libz.dylib

import Foundation

class DownloadManager: NSObject, NSURLSessionDownloadDelegate {
    
    let backgroundSession: NSURLSession!
    
    override init() {
        super.init()
        let config = NSURLSessionConfiguration.backgroundSessionConfigurationWithIdentifier("YOUR.COM.NAME") // Вообще любой уникальный идентификатор, e.g. com.toster.zip.BackgroundSession
        backgroundSession = NSURLSession(configuration: config, delegate: self, delegateQueue: nil)
    }
    
    func downloadWithURL(url: NSURL) {
        let task = backgroundSession.downloadTaskWithURL(url)
        task.resume()
    }
    
    func URLSession(session: NSURLSession, downloadTask: NSURLSessionDownloadTask, didFinishDownloadingToURL location: NSURL) {
        let destination = NSHomeDirectory().stringByAppendingString("/Desktop/unzip/")
        SSZipArchive.unzipFileAtPath(location.path!, toDestination: destination)
    }
    
}

let manager = DownloadManager()
manager.downloadWithURL(NSURL(string: "http://.../data.zip")!)

Распакованный архив будет в /Desktop/unzip/