Hunter gatherers arrived in Gotland around 9,500 years ago, (when the Baltic was apparently more of a lake than a sea,) and stuck around for about 5,000 years–persisting on the island for nearly a millennium (or longer) after farmers had invaded mainland Sweden and displaced the hunter-gatherers there. (Of course, you may note that farmers still haven’t made it to northern Sweden.)

I have not found a whole lot in English about stone-age sites in Gotland, but GotlandsResor–yes, a tourist info page–states that:

Several Stone age settlements are known and many of them has been excavated. Stora Karlsö, Visby, Västergarn and Ajvide south of Klintehamn, also in Ihre and Bjers in the north and finally Suderkvie in the south, which was surrounded by open sea. In the centre of Gotland the oldest settlement, Mölner Gullarve, is located, over 7 000 years old.

According to Wikipedia, Ajvide,

covers an area of 200,000 square metres and was occupied from the Late Mesolithic through to the mid Bronze Age. The majority of the activity on the site took place during the Middle Neolithic period (3100 – 2700 BC). This phase of activity belongs to the Pitted Ware culture. …

The principal feature of the site is a burial ground containing some 80 graves. …

A significant faunal assemblage has been recovered from the site. This suggests that in the late Mesolithic the economy was based upon the hunting of grey, ringed and harp seals, porpoise and fishing. Cattle, sheep, and pigs were introduced at the start of the Neolithic. However, there was a resurgence in seal hunting and fishing by the Middle Neolithic. Cattle and sheep returned during the late Neolithic and Bronze Age.^[1] It has been argued^[5] that the pigs which remain on Gotland during the Pitted Ware phase are in fact wild or feral animals, implying a general return to hunting and gathering during this period and not just a reversion to marine resources.

While northern Sweden has remained agriculture-free due to its harsh, cold climate, the perhaps the apparent abandonment of agriculture and resumption of hunting during the Middle Neolithic was driven by a mild climate creating an abundance of easily-hunted animals–or perhaps we are just dealing with two (or more) separate populations who had their own lifestyles.

Wallin, Wallin and Apel’s Prehistoric lifestyles on Gotland – diachronic and synchronic perspectives adds to our picture of the late neolithic and early Bronze Age:

The settlement pattern from the late Neolithic is unclear, and no settlements with house foundations and distinct cultural layers have been found. … In the Early Bronze age around 1800 BC … the cairns became clearly visible monuments in the landscape. The cairns became the new statement indicating more complex social formations and distinctions that already started in the late Neolithic. The Neolithization with control over land resources and extensive use of domesticated animals was a long struggle during a time period of c. 2000 years that finally around 1800 BC could be put in practice and developed further during the Bronze Age. …

One obvious change which probably indicates the establishment of far reaching contacts is the introduction of metal. Copper started to appear in graves during the late Neolithic … Recent studies of copper in bronze artefacts indicate that southern west Europe is a likely source of origin, and it is almost certain that this alloy found its way to Gotland through bartering/trade.

Wallin, Wallin, and Apel also provide us with some maps:

The Bronze Age appears to have been a good time for Gotland:

The material culture that constitutes the Bronze Age on Gotland is the alloy bronze, large cairns, stone ship settings, rock carvings, cup mark-sites, fire cracked stone mounds and pits.  There are according to the Swedish National Site Survey over thousand cairns on Gotland belonging to the Bronze Age.

“Tjelvar’s grave,” ship cairn, Gotland

It has been difficult to locate distinct settlement areas from the Bronze Age, but … field systems have been found, which have indicated Bronze Age dates. Lindquist suggests that evidences point to the fact that Gotland during the end of the Bronze Age was organised in units that were larger than the extended family level with a possible division of labor into farmers, herdsmen, and craftsmen. During this time was an extensive farming and herding method used. … the land-use changed into intensification of agriculture with arable meadows and grazing in smaller “privatised” established areas with a fencing system, during the pre-Roman Iron Age. These types of smaller irregular farming units are also found in Estonia. Lang calls these “Baltic fields” and according to him they reflect the boundaries of clearing of the arable soil and centered on clearing cairns. Thus they diverge from the larger regular Celtic fields, which reflect a conscious land-division and land ownership. (I have removed the in-line citations for readability; see the original if you want them.)

The local Iron Age began around 500 BC, and is divided into “Pre-Roman,” “Roman,” and “Germanic:”

During the decline of the Roman Empire, an abundance of gold flowed into Scandinavia; there are excellent works in gold from this period. Gold was used to make scabbard mountings and bracteates. After the Western Roman Empire fell, gold became scarce and Scandinavians began to make objects of gilded bronze…

This was, as you know, a time of much Nordo-Germanic movement, and was followed by the Viking Age, which was also a time of much Nordic movement.

In the midst of all this trade (or plunder,) Gotland became one of the most important harbors in the Baltic:

The number of Arab dirhams discovered on the island of Gotland alone is astoundingly high. In the various hoards located around the island, there are more of these silver coins than at any other site in Western Eurasia. The total sum is almost as great as the number that has been unearthed in the entire Muslim world.^[24]

And from GotlandsResor:

Gotland is often referred to as “The World´s Treasury”. Over 145 000 coins have been found in Gotland, a fact that makes the island to one of the worlds most important places in prehistoric finds. … The world´s largest ever found silver treasure dating Viking Age was found on northern Gotland at Spillings in 1999. It weight over 80 kilos!

The Spillings Hoard is truly remarkable:

The silver hoard consisted of two parts with a total weight of 67 kg (148 lb) before conservation and consisted of, among other things, 14,295 coins most of which were Islamic from other countries. A third deposition containing over 20 kg (44 lb) of bronze scrap-metal was also found. … As of 2015, more than 1,000 kilograms (2,200 lb) silver from over 700 caches deposited between the 9th and 12th centuries have been found on Gotland. This includes 168,000 silver coins from the Arab world, North Africa and Central Asia.^[16]

Map of the Hanseatic League

Gotland went on to become an important trade point in the Hanseatic League, (1400-1800):

Visby [Gotland] functioned as the leading centre in the Baltic before the Hansa. Sailing east, Visby merchants established a trading post at Novgorod called Gutagard (also known as Gotenhof) in 1080.^[2] Merchants from northern Germany also stayed in the early period of the Gotlander settlement.

Gotland’s Visby was, for a time, the second-most important city in the Hanseatic league, until the Danes decided to conquer and loot it, prompting a war between the Hanseatic League and Denmark. Denmark lost, but kept Visby (and all of Gotland.) From there, it degenerated into a pirates’ nest, and in 1470 was soon stripped of its Hanseatic membership.

Still, not a bad run–from back-water hunter-gatherer hold-outs to one of the wealthiest islands in the world in just a few thousand years.