Is functional GUI programming possible?

My question is, is it possible to have a functional approach to GUI programming?

The key words you are looking for are "functional reactive programming" (FRP).

Conal Elliott and some others have made a bit of a cottage industry out of trying to find the right abstraction for FRP. There are several implementations of FRP concepts in Haskell.

You might consider starting with Conal's most recent "Push-Pull Functional Reactive Programming" paper, but there are several other (older) implementations, some linked from the haskell.org site. Conal has a knack for covering the entire domain, and his paper can be read without reference to what came before.

To get a feel for how this approach can be used for GUI development, you might want to look at Fudgets, which while it is getting a bit long in the tooth these days, being designed in the mid 90s, does present a solid FRP approach to GUI design.

The Haskell approach seems to be to just wrap imperative GUI toolkits (such as GTK+ or wxWidgets) and to use "do" blocks to simulate an imperative style

That's not really the "Haskell approach" -- that's just how you bind to imperative GUI toolkits most directly -- via an imperative interface. Haskell just happens to have fairly prominent bindings.

There are several moderately mature, or more experimental purely functional/declarative approaches to GUIs, mostly in Haskell, and primarily using functional reactive programming.

Some examples are:

• reflex-platform, https://github.com/reflex-frp/reflex-platform
• grapefruit, http://hackage.haskell.org/package/grapefruit-ui-gtk
• reactive, http://hackage.haskell.org/package/reactive-glut
• wxFruit, http://hackage.haskell.org/package/wxFruit
• reactive-banana, http://hackage.haskell.org/package/reactive-banana

For those of you not familiar with Haskell, Flapjax, http://www.flapjax-lang.org/ is an implementation of functional reactive programming on top of JavaScript.