WebSites that use Design Pattern Framework 4.0

I bet it will be very difficult to find one.  The simple reason is Design Pattern Framework is a collection of many patterns and not all application requires all the patterns.

There may be elements of some patterns from the framework around, but nothing in full..

I hope somebody could prove me wrong.  Rather I would say, blindly taking a framework and developer your applications (business applications) won't be a good idea, unless you have done enough research on the framework, and decided, yes, everything fits the business need..

Because sometimes fully patternizing your solution, at the end may make it more unmaintainable. 

Sprinkle patterns as and when required in your application and you will be good to go.  There are some areas, where you can apply some pattern proactively, but not always.

It also depends on the scope of the application you are targetting.

So, in a nutshell, atleast in the public domain you won't find a full implementation apart from some POC code.  In enterprise applications yes most of the patterns are found in one form or other, but may not be directly.