Does any buddhist school, extant or otherwise, say that there is no svabhava what-so-ever? Does any buddhist school, extant or otherwise, say that there is no svabhava what-so-ever?

I was thinking maybe an early school without the abhidahrma (mahasanghika) or perhaps prasangika-yogaraca in which non duality and convention / absolute emptiness is all there is.

However, do these schools say that there is no essence?
I will explain some views in accordance with Madhyamaka-Prasangika:

"Essence" is something assigned by mental elaboration.

Therefore we can say that apart from mental elaborations there is no essence.

Another view is that

whatever appears (as an individual phenomenon) is its own essence.

That's because the phenomenon is what can be directly perceived; anything else would be mental additions (conceptualizations, associations). Therefore only the phenomenon itself can be its essence.

In that sense we can say that appearance and essence (or "essence and function") are the same thing.

See also discussions on svalakshana.

Note that, in order to understand "what is", we can't rely on flat view of "is" or "isn't". We should see how does appearance relate to mental processes; only there we can see real answers.

So "existence" can be seen as a construction, a way we construct our world - elaborating over directly perceived appearances.

And the deeper we go there, the more we understand that ultimately everything is "Don't Know".

PS: Note that from other points of view other concepts will be true. It would be wrong to think that a Buddhist school claims its concept as the only possible truth. For example, in Zen we can use views both of Yogachara and Madhyamaka, though they seem to contradict. "Everything is Consciousness" and "Consciousness doesn't exist". Logically these two sentences seem to contradict, but in fact they can be seen as both true; only as constructions built from different starting points; describing different levels of analysis. Likewise, we can discuss the duality of essence and function - in order to understand something - and then we can say that essence and function are the same thing, and understand something even deeper.