My app should download small piece of data from network every day (or week/month/etc, that is configurable) and save it to local file. Is that possible in iOS? Is it possible to run app in background mode, download data and stop it?
It's possible only if your app has background mode enabled and it's in the background state. Background state is reached when user exit from your application and until garbage collector doesn't kill it for memory usage.
If you're goal is to download something every day / week / month in background even if user doesn't use app, that's impossible on iOS.