Definitely made up for different reasons:
1) The cancellation stops at the edge of the stamp. It has to be taken from another letter.
2) If it was sent TO
Iceland, why is there an "Island"-stamp on it?
3) At best it was definitely sent from one town on the island to another. So why did they add "Country: Iceland" in the address?
4) In Búđardalur are living not more than 300 people, and there is just one Shopping "Center". So why not take a phone book and call them for the name of the lady? Should have been no problem at all.
5) I took a look at google maps and compared it with the drawing. There is only one single farm, which easily can be found -> https://holar-farm.business.site/
So a very nice try, but no way this letter was ever sent!
Excuse my time clauses/conditional mistakes, I'm Austrian.