Adding to this: genetic evidence shows that Icelandics, like Shetlands and Fjords, are pretty far removed from Spanish/Arabian blood. They're closer to Przewalski's horses and (I presume) Mongolian ponies. The below image is from Lucy Allen and Wallner et al (2017)'s works.
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Then Prystupa (2011) found that: 1) Icelandics are closer related to Mongolian ponies than Norwegian Fjords, 2) the rare E haplogroup (only 3% of the world's equine population have it) was the most common haplogroup in the Eriskay, Dales, and Kerry Bog, and also quite common in the Shetland, Icelandic and Fjord (35%, 8%, and 8% respectively), 3) today's E haplogroup dates back as far as the Bronze age in Siberia.
You can find these studies on the Eriskay Pony Society website - I very much recommend a read.