Is there, or has there ever been, any jurisdiction governed only under common law, with no statutory law?
Yes. The first English statute was enacted in 1235:
The Statute of Merton or Provisions of Merton (Latin: Provisiones de Merton, or Stat. Merton), sometimes also known as the Ancient Statute of Merton, was a statute passed by the Parliament of England in 1235 during the reign of Henry III. It is considered to be the first English statute, and is printed as the first statute in The Statutes of the Realm.
Common law was established in the previous century:
In 1154, Henry II became the first Plantagenet king. Among many achievements, Henry institutionalized common law by creating a unified system of law "common" to the country through incorporating and elevating local custom to the national, ending local control and peculiarities, eliminating arbitrary remedies and reinstating a jury system—citizens sworn on oath to investigate reliable criminal accusations and civil claims.
Therefore, from some time in Henry II's reign until 1235, England was subject to common law while having no statutory law.
The problem with the question is, that there is clearly a misunderstanding with what is the history of what we call Common law today. Common law is not a concept that applies to all laws, it'S one legal school that developed distinctly and incorporated customary law. However, in a silly quirk of history, the very moment that we actually can speak of the Common Law as a legal system, all its foundational laws had been written down and ordered to be the law of the land, making the very foundation of Common Law the statutory codification of the customary law.
What is statutory law?
Law is statutory if it is written down in statutes, so any law that is passed by any body of legislature, then written down and then distributed. A couple examples are the Code of Hammurabi - predating any common law we know today - as well as the 12 Tables of Rome or the Solonian Constitution - which is a poem.
What is Common Law?
The common law is a specific type or school of legal system where precedent generates binding law. De facto we speak of Common Law only since Henry I or Henry II, so at the earliest about 1100, but more typically we just assume the 12th century to be the time when the term is applicable. It developed from customary law, but is not one.
What is customary law?
Customary Law is much looser defined than statutory law. A customary law exists where
- a certain legal practice is observed and
- the relevant actors consider it to be an opinion of law or necessity
However, unlike other laws, customary law is usually not codified in the same way as other laws, though it may be written down. It is not a paradox to have first codified and then statutory customary laws, though that is the last stage of customary law usually. Often, the next steps are law reforms where the legislative body adjusts them to reflect the current reality. That influence of the legislative body is removing the part that makes it customary.
England had Statutory Law before it was Common Law!
When the Normans invaded under William the Bastard in 1066, they came also armed with their norman legal system, which had been developed from Frankian and Norse Law (more on that later). The Normans had the "Magnum Concilium" - or in English: great council - which would develop laws or render judgments. Especially the judgments would be written often. Since their landing in France, the Normans had assimilated a lot of Frankian law, which was the Salic Law and statutory since about 500 AD. However, the Norman legal tradition was to a good degree customary law, despite incorporating some of Salic law and canon law where the church was involved.
Technically, the Doomsday Book is also a law by writing down who owned which land, making it statutory, but it is more akin to a legal document that gave land and rights. Those are called charters and the Normans started doing this around 1050.
Norman Rulers of England
After the conquest of Britain in 1066 and two inheritances of the crown later, Henry I, the fourth son of William the Bastard, ordered all the customary laws written down by royal decree. This happened around 1110, making the English laws that would within a generation be turning into Common Law statutory by definition: The legislative body - the king - ordered the customary law of the land written down, so it is statutory.
Only Henry II (1154-1189) would create the wide-sweeping legal reforms that would later lead to what we understand as and call Common Law today. Those revisions include the establishment of the circuit judge and the formalization of Jury trials.
This even predates the English parliament: Only after being crowned in 1216, Henry III called his great councils Parliament and had them sign off on tax laws based on the Magna Carta of 1215 - but those assemblies were unlike modern parliament. They were pretty much a royal assembly, as in both a court of law, but also a meeting of the lords and barons to plea for things in the royal court - which wasn't just a court of law. It'd be more prudent to call the Parliament at that time was very much the legislative, judicative and executive power all assembled in one place, and the highest officer of all three was the king.
Pre Norman Conquest
So, what about the time before Henry I? Is there Common Law without statutory law? Well... let's see... No, technically a lot of England couldn't qualify as un-statutory after 42 AD because England started to become subject to the very well-written down Roman law then. Roman law was statutory since the 5th century BC, the Twelve Tables. Then the church in England had of course canon law, starting around 400 AD.
Now, the kicker is, that one of the oldest surviving English law treaties was the Law of Æthelberht. It was written in the 7th Century AD, predating Common Law by more than 400 years! The name-giving ruler died in 616 AD, so make it about 500 years.
Many of the customary laws that would be around in England around 1100 could trace their lineage to those, or the laws of the Dane.
On a technicality the Danelag does not qualify.
The common law would incorporate a big area in which people were using Danelag - or Danelaw, law of the Dane people.
When the Norse people invaded England after the Raid of Lindosfarne and the conquer of York, they brought their own legal system with them. This was at least at landing - unlike for example Mercian law - not influenced by Roman law (statutory), canon law (also statutory since 325), and local customary law. As pointed out above, the local customary law had in part developed from those two and older. To confuse everything, historians call the combined lands owned by the Danes, Swedes, and Norwegians in half a dozen to dozen separate lordships collectively the Danelaw - even though they had no combined king or Warlord. To separate the land from the law, I'll use Danelaw for the territory, and Danelag for their legal system.
The legal system in the Danelaw (865-954), and North Sea Empire (1013-1042), as well as many other countries within the Norse sphere, was pretty much the Danelag, the law of the Danes, and lo and behold: It was only codified in the early 11th century as Dena Lage. In fact, Medieval Scandinavian Law, which the Danelag was a subset of, used an oral tradition of lawspeakers. These, mostly older men, would memorize all laws and hand them down together with precedent and commentary. When the laws were needed, they were recited in the appropriate setting. So the Danelag was not statutory but an oral customary law, though decrees of rulers (be they a lord or the council of free men on a thing) had also the power to change laws or invent new ones. Only at times, some of these decrees were put into writing.
However, the Danelag is clearly not common law, since that term can only applied to the English legal system after Henry II, who used the statutory writeups Henry I had ordered to unify and reform the legal system into what we recognize as common law today.