Can a No-Knowledge Proof exist?
No.  An algorithm that has access to some information can always ignore it and make decisions without looking at it, and then its externally observable behavior will be indistinguishable from an algorithm that doesn't have access to that information; so no verifier can tell the two apart.
I suggest being a bit cautious: when we talk about words like &quot;knowledge&quot; in the context of algorithm, we are applying some human intuition and interpretation to the underlying mathematics.  The precise meaning is given by the formal mathematical definition.
Incidentally, it's called a proof of knowledge, not a knowledge proof.