I was able to export the zone file manually by following: https://www.godaddy.com/help/export-my-domains-zone-file-records-4166
Is there any way to do this programmatically, using API keys. This is necessary for backup to periodically take a export of the zone file.
We don't directly support an API key for our DNS Manager. However, I believe there was another discussion with members who worked out a solution for something similar to what you're requesting here. I hope this helps.