In New Zealand - there is one, at least according to government highways agency Transit New Zealand which states on its site:
"The Arahura River is unique as it is the only privately owned river in New Zealand"
In case you didn't know, the Arahura river is on the South Island's West Coast. Transit's only interest is that it is responsible for the single lane State Highway 6 bridge over the river, which it shares with OnTrack - as the branch railway line between Greymouth and Hokitika shares the bridge with the road, causing a few headaches for motorists when they have to give way to trains. Transit has some of your petrol tax money to investigate options for replacing the bridge, but that is not my issue and there is no claim that the fact the river is privately owned is causing any difficulties with this project.
What IS important is that, if true, the Arahura River is an example of what can be done with other rivers. The world has not fallen in, Grey District Council has not foretold disaster and nobody seems to notice. A bit of research uncovered that Mawhera Incorporation owns the river according to Trade and Enterprise NZ. In essence, a company owned by a local Runaka (subset of Ngai Tahu). Nothing wrong with that. I would presume the Maori Party would support this being maintained, as does the Libertarianz, as should ACT.
It is one response to concern about libertarians privatising what is seen as "the commons". Would anyone notice if all the rivers were privately owned? Who would want to nationalise it?