@Stats, @mlofton, and @asdf are all correct. Let me try to summarize. 

Non-stationarity is not a single concept. Rather it is the lack of a single concept -- stationarity. This means that there are (as @Stats mentions) an infinite number of non-stationary structures. There is a non-stationary model where the truth is an ARIMA model (with your specified p,d, and q), but where the coefficients associated with the p and q change every 10 periods. Could the ARIMA model do a good job of modeling that? Making predictions in that setting? Learning coefficients under those conditions? No. 

But there are some non-stationary structures the ARIMA model can handle. An $AR(1)$ model is non-stationary in one sense -- the distribution of each time period is dependent on the previous one. But for data which is truly $AR(1)$, we can model it (under some conditions). Similarly, data which follows the ARIMA structure is inherently non-stationary. However as long as it sticks to that type of non-stationarity -- namely some finite number of differences being modeled by an ARMA process  -- things are manageable.

This is why there is some disagreement. It is true that there are types of non-stationarity that ARIMA models can handle, but it is also true that there are many types that it is totally worthless in the face of.