To answer your question straight away: No, Iceland is not a continent. But we can see why you would think that when you look at a map. You might also ask yourself what continent Iceland is on. That requires a more complicated answer.
Iceland is not on one continent but two!
If we dust off our geology, you might remember that the world sits on tectonic plates. Most countries are just on one plate, but the thing is, Iceland is on two. The North American plate and the Eurasian plate. That means Iceland is technically in two continents. The fact that Iceland stands on plate boundaries also explains why there is so much volcanic activity.
Is Iceland in North America or Europe?
Although theoretically, we could split the country to be legally in both North America and Europe, Iceland is considered to be in Europe. Iceland is part of the EEA (European Economic Area), and since most of the island is on the Eurasian plate, it makes sense not to divide the nation to be in two continents legally. No Icelander argues for it, and we all consider ourselves to be European. This is mostly because in the olden days almost all of our foreign contact was to European countries.
You can see where the plates split
There is a visible crack that runs through the whole island. And you can see it for yourself. The most popular place to see the split is definitely Þingvellir national park. If you take the Golden circle route, you are certainly going to view it. In Almannagjá in Þingvellir you are actually walking down the split and are between continents. Another, not as famous, place to see the crack is at the Bridge between continents at Sandvík. There you can walk over a bridge and be between two tectonic plates. It’ a popular photo spot. You can also go down to the crack and walk along with it.