Fragaria vesca; the Wild Strawberry is more or less common in warmer regions of Iceland, like the south but also in the north on sunny spots of south-facing slopes. Where conditions are not so favourable fruits don't ripen fully. Without flowers or fruits they can easily be confused with the stone bramble. However, one can tell them apart anytime by the top-leaflet. Brambles have clearly stalked top-leaflet where strawberries have sessile top-leaflets.. The wild strawberry has white flowers in summer and red "fruits" in late summer/autumn. It might be interesting to know that a strawberry fruit is'nt really a fruit but a swollen juicy flowerbase. The real fruits are the single kernels on the berry.
It is a member of the Rose family: Rosaceae. The Icelandic name of this species is Jarðarber.