The Files application, added to iOS 11, supports zip files. You can open them, view their contents, and extract files without any third-party applications. If you want to create zip files, you will still need an App Store application.
When you download a zip file in Safari, your iPhone or iPad offers to open it the Files application. Tap "Open in files" to do it. You can also save zip files in the Files application from other applications.
You will be prompted to choose a location for the zip file. You save a copy of the zip file to this location.
Select a location, such as your iCloud player or a folder on your iPhone, and tap "Add."
Tap the zip file in your Files application to open it.
Tap "Preview Content" to see the contents of the zip file.
You can see the contents of the zip file here. If the zip file contains multiple files, swipe left and right to switch between them.
To extract a file from the zip archive, press the Share button and select "Save to files" to save it to your file application or tap an application to send it immediately to this application.
If you want more features (for example, if you really want to create zip files on your iPhone or add files to an existing zip file), you'll need to a third-party application to work with zip files.