There is a river in Reykjavik, the Elliðaár. It may not be navigable, and it doesn't go through the city center, but it is a river, and it flows within the city boundaries.
On the other Hand Irish Monks (Papar), like the holy Brendan, probably discovered Iceland before the Scandinavian Settlement and may have even been the first settlers in Iceland. But of course that should be seen with strong doubt, because there is no archaelogical Evidence for that.
The vikings might not have been the first to inhabit Iceland. The sagas describe the island as having been already inhabited by Papars, a celtic monastic people, before the arrival of the first norse settlers. Since there is zero evidence for the existence of Papars outside of the sagas, the norse might have been the first settlers, but the sagas might also be truthful and celtic monks may have colonized the island some time before the norse showed up.
I knew it wasn't just me when I saw it read "per capital" instead of "per capita"