I think this is the case. When I lived in Minneapolis (Target's HQ), Target were in abundance and in competition with each other, so sometimes one store would have higher prices than another.
That's actually not true. There are three national price tiers that Target uses. There is a base national price, a lower price for proximity to Walmart stores (I don't know how close they have to be), and a set of testing stores where they mark items at a higher price then compare sales rates to the national base price stores. So you might see a base price of $10.99, a WM competitive price of $9.99, and a price test store with a price of $11.99 for stores all in one market. That said, I don't think individual stores ever manually set their own pricing just to compete with the Target down the street, though it is obviously possible to override pricing for an individual store as we saw with the legendary Knollwood sale.