Sure but the UK is not Switzerland or Iceland.
On the world stage Iceland is irrelevant. It's a lot like Switzerland without influence. Sure, Iceland can afford to be eccentric about world politics because it's not under the direct influence of other countries to the same extent as most are but it also doesn't hold any real influence itself. The UK doesn't want to be Iceland, it wants to be like Iceland but with the political weight of the US.
Switzerland is more relevant, especially financially, but Switzerland is in a unique position that is the result of centuries of European history and of course its location and geography. The UK is just a bunch of islands off the coast of France that used to be a naval superpower (like Portugal). The UK can't be Switzerland because it isn't already Switzerland -- Switzerland is Switzerland.
If anything, Brexit was fuelled by impotent rage: the UK used to be an Empire, now her citizens are supposed to bow to a council of European nations and become just another cog in the machine of the continent. We want our Empire back! We're important! We shouldn't be subject to Europe, we should compete with it directly.
Except the UK doesn't matter anymore either. The UK used to matter up to WW 2 where it exhausted its remaining military might to help keep the free world free (which btw as a German I fully acknowledge and am thankful for). But even then it had already lost (or was in the process of losing) its colonies and was largely confined to its homeland and some assorted junk territories that ultimately don't matter much (sorry, Falkland islands).
Geographically, the UK needs to establish a working relationship with the EU to survive. Before Brexit it had it all: it was a member of the EU but had been able to negotiate special statuses all along the way by threatening to leave. Now it basically has to start from scratch and has only its own worth to prop itself up during the upcoming 2 years of bartering.
And that worth isn't much, to be honest. Sure, London is one of the most important financial centres of Europe. But in a large part it only stayed that way because Britain heavily relied on special exemptions from EU rules while still drawing on the benefits of that market. Sure the UK now definitely gets to keep its currency, but it no longer competes with Frankfurt but with New York, Tokyo and Shanghai. It's not a big fish in a pond anymore. And the pond has no more reason to be impressed. Economically the UK relies heavily on extremely expensive imports (it's a bunch of islands after all) and those are only going to become more expensive after leaving the EU. The financial centre is the biggest asset and its importance is heavily impacted by Brexit. Things are looking grim.
So in other words: the UK won't be the next Switzerland. It may become the next Iceland. But only in the sense that it will be out in the sea off the shore of France, doing its own thing while occasionally pouting about how silly everyone is and nobody will pay any attention.
At least this is the course Brexit seems to have taken the UK on. And if Scotland makes true on its threats to secede from the union, the UK may just end up as the good old England and Wales all alone, surrounded by Europe. But on the plus side Wales may finally have some pressure to get the UK to put that dragon on the flag.