One could argue that ancient Egypt had the longest continuous run of sovereignty, starting with the First Dynasty around 3100 B.C. and running to 343 B.C., when it was conquered by the Persians. That's about 2700 years, give or take a few, unless you count the Hyksos invasion in around 1650 B.C. as breaking that string. It's likely that only one modern nation -- China -- can trace its origins back 2700 years or more, but it was conquered in the thirteenth century by the Mongols.
This is the kind of question, however, that likely has no satisfactory answer. The notion of "sovereignty" is a relatively recent innovation in history. Prior to about the eighteenth century, the idea that a "country" was anything but the domain of a sovereign would have been regarded with some suspicion.
Furthermore, it isn't very easy to pinpoint the beginnings of many nations. When, for instance, did France first exist? Was it when the Franks kicked out the Romans in the fifth century? Or was it in 843 when the Treaty of Verdun split the Carolingian empire into three parts and made Charles the Bald the king of the western third that ultimately formed the basis for France? Or was it some other date?