Let $\ C\ $ be an uncountable compact subset of $\ \mathbf R.\ $ Let $\ K\ $ be the linearly ordered set obtained from $\ C\ $ by identifying every two $\ a\ b\ \in\ C\ $ such that $\ (a;b)\cap C=\emptyset.\ $ (Sometimes more than two points can get identified together into one class but this is fine). Set $\ K\ $ is still uncountable (all equivalences classes but a countable number consist of exactly one element). The topological space canonically associated with $\ K\ $ has the minimal and maximal point, and it is easy to check that $\ K\ $ has Dedekind property (for an interval) induced by Dedekind property of $\ \mathbf R.\ $ Thus $\ K\ $ is a connected Hausdorff compact space.

(There is a Urysohn function of $\ K\ $ onto a non-trivial interval).