Is there any open-source (free to use) code available that provides a vertical page index with tab-bar like functionality. The UI is usually inspired by paper based phone diaries and/or note books.
For example, see attached screenshot of Daily Notes + Todo iPad application. I’m actually looking for something more like the tabs in Chrome (which are actually overlapping).

Note: I’m looking for a solution that works with UISplitViewController (detail view, in particular). And, in my case each “button” will overlap (just like in actual paper based diaries indexes)

Not to my knowledge, I’m afraid. You can check out a repository of many UI frameworks at cocoacontrols.com
To be honest, this wouldn’t be that hard to implement yourself. And you could the open source your own work and provide it for others to use.