“As they step into the same rivers, other and still other waters flow upon them.” (Cp. S, R, DK 12)
Compare L&M D65b: “It is always different waters that flow toward those who step into the same waters.”
Heraclitus has a few statements that speak of the flux of rivers. As a whole they don’t support the view of him as a philosopher of radical non-identity that has come down to us.
One interpretation of the present fragment is that it is a definitional statement, like many others that we find in Heraclitus. Rivers, he is saying, are the kind of things that have ever-flowing waters. In this way they differ from ponds, for example, which we (at least in everyday folk views) understand as bodies of water that are not ever-changing.
In any case, he is speaking of rivers here as permanent things insofar as we can continually return to them. We go again and again to the Rhine or the Mississipi, or the Cayster river (an important river in Ancient Ephesus that Heraclitus would have regularly seen). The waters of the river will be ever new, but the river still maintains its identity.
Yet when Heraclitus speaks of rivers, he isn’t merely speaking of rivers. This fragment is from a text in which Cleanthes is comparing the views of the soul of Heraclitus and Zeno. Cleanthes, like Plato and others, view Heraclitus’ comments about the river as statements about reality more generally.
Kahn quotes what he takes to be a Heraclitian insight expressed in Plato’s Symposium (thought to be composed, by the way, by the Athenian river Ilisos), applying what Heraclitus says here to the personal identity and even the identity of the soul:
Mortal nature seeks…to be forever and to be immortal. But it can only do so by…leaving something else new behind in place of the old…as when a man is called the same from childhood to old age. He is called the same despite the fact the he does not have the same hair and flesh and bones and blood and all the body, but loses them and is always becoming new. And similarly for the soul: his dispositions and habits, opinions, desires, pleasures, pains, fears, none of these remains the same, but some are coming-to-be, others are lost. (Symposium 207D)
Plato, like Heraclitus, is drawing attention to change. Yet while they are both in these passages emphasizing process, neither of them — certainly not Plato! — maintain a radical doctrine of no-self and non-identity. More to this in the commentary of Fragment 51.