Is it possible to have an AR(0) for streamflow? Autoregressive models (AR) are extensively used to understand the behavior of streamflow. The literature abounds with examples of AR models from different families applied "everywhere" to fully understand how streamflow changes throughout the hydrologic year. One particularly well accepted model is the periodic autoregressive model, as in Hipel and McLeod (1984).

So far, from all I've ever read, all streamflow AR models will have a signficant first order lag. That is, when one calculates the PACF to find the order p of the model, p will assume a value of 1, at least. 

Is there any known case registered in the literature of an AR streamflow model of order 0, i.e., AR(0) for streamflow? That is, is there a possible way that the "yesterday's flow" won't explain the "today's flow"?
Autoregressive models are mathematical models, and like all models, are based on a conceptual model of the system. The conceptual model is that prior flow is a parameter in prediction of current flow. So it doesn't really make sense to have a model where it does not do so. 

Furthermore it is difficult to conceive of a natural system where prior flow is not a predictor. This is where I disagree with Gordon's answer. Even in small catchments, prior flow is a good surrogate for the physical conditions that contribute to flow. The conditions include precipitation history, soil moisture, filling of depressions so flow can accumulate, capacity for trees to intercept precipitation (are the leaves already wet), water-table stream connectivity, etc.

If you don't see a relationship to prior flow in small catchments, that may just mean your daily timestep is too large. I care about the flow variation in very small catchments and monitor the flow on 30-minute intervals.