Is it possible to program an iOS app using an iPad?

Yes and no, but mainly no. Let me explain.

You can't use an iPad (at least not yet) to go through the entire process (i.e. design, test, and submit your app). You also can't use Xcode on an iPad (at least not directly, although you could do this remotely - but that is not a practical solution).

People can and do use iPads for:

• learning how to code
• editing code using various editors (but not Xcode)
• designing some of the other resources/assets used in apps (e.g. images, icons, sounds, etc)

However, most development is done using an integrated development environment (IDE) such as Xcode. Most apps for iOS (and macOS, tvOS, and watchOS) are programmed and submitted via Xcode, although there are alternatives.

Codea is an iPad app that can be used to develop mainly games for the iPad. You can develop and test your game on the iPad but if you wish to publish it you will need to go thru Xcode. Codea uses Lua as its programming language. Codea is quite popular and has an active helpful internet message board.

I've also heard of people writing apps in Python. There is an iPad/iPhone app, called Pythonista, that allows you to write and run python scripts on an iPad and iPhones.

Starting with iPadOS 15, Apple is adding the ability to "Build apps on your iPad" (WWDC June 7th 2021 video timestamp 47:26). According Apple, you will be able to build, preview and submit apps for iPhone and iPad to the app store from your iPad when iPadOS 15 is released this fall.

If you want to make iOS app using Xcode then probably not. You can use iPad for learning Swift programming language using Apple's Swift Playgrounds app though.