An example of a real-world map that is not 4-colourable?

Wikipedia has a list of sets of four countries that border one another; it includes a few divided countries (Russia, in Belarus-Lithuania-Russia-Poland, with Kaliningrad being separated from the rest of Russia; and Azerbaijan in Azerbaijan-Iran-Armenia-Turkey). There do not seem to be any set of 5 or more currently.

Added: However, one could argue that the existence of disconnected countries in fact forces certain maps to use more colors, even though they do not involve disconnected countries. For example, a map of Europe would necessarily require different colors for France, Belgium, the Netherlands, Germany, Denmark, Norway, and the United Kingdom (possibly also Ireland), lest it seem that one of those countries is disconnected into two non-adjacent regions. This also shows some of the difficulties that islands produce (as opposed to the idealized maps from the theorem).

There was a time when Britain, Spain, France, Portugal, and the Dutch Republic bordered each other (early 18th century, counting borders lying between colonies); and from the late 19th century until before World War I, so did Belgium, Portugal, Germany, France, the United Kingdom, and the Netherlands (necessitating six colors; again, this includes colonies that may border one another).

It's possible that some smaller divisions (states, provinces, etc.) might include such sets.

As I noted in the comment, there is an incorrect assumption in your question that a counterexample must have disconnected regions: in the four-color map theorem it is assumed that neighboring regions have a border of positive measure. Countries/states meeting at a point (such as Colorado, Utah, New Mexico, and Arizona meeting at the Four State Corner) are required to use different colors in maps, but in the context of the 4-color map theorem regions that only meet at the point could be colored using the same color.

Here's a map of the Holy Roman Empire after the Peace of Westphalia in 1648.

In the color scheme the map makers have chosen, the pieces of the map labeled "Bavaria" (green), "Austrian Hapsburg" (light green), "Ecclesiastical" (light purple), "Hohenzollern Franconian" (light brown, including Ansbach and Bayreuth), and "Imperial Cities" (red) all touch each other. So five colors are required to differentiate them.