Although neither Apple's Safari nor Google's Chrome browser can extract files from ZIP archives on iOS, it has been discovered that Apple's Notes app lets users view the contents of ZIP files by taking advantage of iOS's built-in file attachment viewers.
Starting with iOS 9, Notes has received several new features, among them support for adding content to a new or existing note through iOS's multi-purpose Share menu in apps like Safari, Maps, Pages and more.
As it turns out, you can also use iOS's Share menu in conjunction with the “Open In” feature to add a downloaded ZIP file to a note on your iPhone, iPod touch or iPad.