Are there any offline applications that support or work with Trello?
Also, is there any offline tool that performs a similar function (kanban) ?
You can use Kanbanik for your Kanban requirements. It can be installed locally and accessed via a suitable web browser.
Documentation and source code is available via the project's GitHub repository.
There are 3 applications I could find:
or you can use your evernote as kanban with:
Also feel free to vote on trello for offline support.