Hello,
As you know, K2 does not support a dependent extra fields, so by default you can not get form fields to work like that.
But JA K2 filter component provides a feature called “Dynamic counter” which helps to calculate item number that matches with each search option based on what user selects, and disable (or hide) options that have no item.
You can get more information about this feature on our documentation page here
http://www.joomlart.com/documentation/joomla-component/ja-k2-filter#version-107
Feature 2: update item counter for each value of form field depend on what user selected
So, how this feature can apply for your case?
If you can update your K2 items with reference data (select correct country and city for each items), then on search form, if you select the first level select box, then the options in next level select boxes will be re-calculated depending on what you selected, and options that have no items will be disabled/hided and un-selectable.
Please visit our demo page for a closer look:
http://joomla-extensions.demo.joomlart.com/index.php/ja-k2-filter/dynamic-counter
P/S: you need add some custom javascript code to show/hide city selection box depend on value of country box.