If you accept a certain degree of fuzziness as to what constitutes 'the government,' the Japanese monarchy was traditionally started in the 7th century BC. If you want to talk more local/non-sovereign, the area of the modern city of Faiyum in Egypt has been continuously inhabited since the 4th millennium BC. I don't know about continuity of government, though. Finding that out would be a good deal harder and would probably require a team doing dedicated research, since local records are often more spotty and less well maintained than, for instance, royal archives.
I don't think your question has a simple answer; the exact definition of "continuous" matters a lot. Was the English government "continuous" through the Civil War or not?
One interesting case to consider is the Roman Catholic church. It's gone through quite a lot over the years, but has always ruled over some territory and been fairly continuous. There have been multiple claimants to the papacy on several occasions, but this was always settled, leaving one line as "legitimate".
The two we sort of know for certain are the Tynwald on the Isle of Man, which celebrated its millennium anniversary in 1979 (albeit, there isn't too much supporting evidence behind 979 as the start date) and the Faroe Islands Alting (literally "Law Thing") which is thought to be older, although, AFAIK, we don't have an exact date only it is thought to be 9th Century. The Icelandic parliament is also slightly older than the Tynwald but had a gap of about 25 years in the 19th century, so it can't really be counted because of that.
A vague question, I'll try to answer as best as I can. Officially recognized states will be much more recent than historic governments, because there was no international framework in place to recognize said states until the formation of the League of Nations Ginneken
Technically the answer to your question is the United States of America. No other country has currently held the same government for a longer period of time. However, if we're willing to be a little lenient, my answer would be Japan.
Japan, which has had a continuous (ish) government since around 510 AD, which is dozens of generations into their proclaimed history, but we've yet to find solid history to back up claims before that time frame. Since China has changed emperors, dynasties and even ruling ethnicities multiple times, I'll take a firm stance and say they have not had continuous government, which goes for Iran and Egypt as well. Japan however, has had the same government head, the emperor, until it's demise in 1945. Since the Western nations had at that point already established an interim government under MacArthur in Japan before making the emperor announce nationwide that he is no longer government head, we could say that they have the oldest continuous government.
San Marino, while claiming sovereignty since 300 AD, doesn't actually have hard evidence to back up their claims until after 1000 AD. Not to mention that this rule is not continuous, because for a period of 10 or 20 years in the 30s, San Marino was taken over by fascists, which disqualifies it. If it wasn't for this twenty year gap, it would hold the record, because it's Constitution is older than that of the US
Not sure if it's the oldest government in the world, but some Dutch Water Boards (Hoogheemraadschappen) have operated since the 13th century. This is a regional government body.