Did the Buddha ever refute the view of "extreme metaphysical nihilism" e.g. a view that "nothing is real"? Important note: I am only using the word "real" only in the sense of meaning "exists and is not an illusion". 

I am defining "extreme metaphysical nihilism" as the belief that "nothing is real". 

I do not know if the belief that "nothing is real" is (also) called something else but if it is then please tell me. 

I consider the view that "nothing is real" as being false. 

A refutation of the view that "nothing is real" is that consciousness is observably real in the ability to understand "real or unreal". Mental suffering may be verified as real by observing that it exists in reality as what it is deemed to be. 

Did the Buddha ever refute the view of "extreme metaphysical nihilism" i.e. a view that "nothing is real"?
The Lump of Foam Sutta  states:


  Form is like a lump of foam,
  Feeling like a water bubble;
  Perception is like a mirage,
  Volitions like a plantain trunk,
  And consciousness like an illusion,
  So explained the Kinsman of the Sun.
  
  “However one may ponder it.
  And carefully investigate it,
  It appears but hollow and void
  When one views it carefully.


Some people use this sutta to claim that nothing is real. However, the commentaries state that this refers to the five aggregates (form, feeling, perception, mental formations and consciousness) being empty of a self. That's consistent with the Empty Sutta. The sutta also implies that the five aggregates are impermanent, conditioned and changing.

From Piya Tan's commentary on this sutta, for which he refers to other sources also:


  2.1.1 Just as a lump of froth is unsubstantial, so too form is 
  unsubstantial due to its lack of any abiding entity, permanent
  substance or self-essence. The froth cannot be used to make such
  physical things as  a bowl or a saucer, but it merely breaks up.  Even
  so, form cannot be taken as permanent or lasting, or as “I” or as
  “mine.” It is merely a lump of froth, impermanent, unsatisfactory, not
  self and foul.
  
  2.1.4 But a lump of froth ever breaks up as soon as it has arisen, and even if it lasts for a while, on reaching the sea, it always breaks
  up. Even so, this body is continually breaking up and changing, but
  breaks up finally within a century, and after death, it disintegrates
  into tiny fragments.


Commentary on the other metaphors can be read from the link.

This means that the self is ultimately an illusion, and an emergent phenomena. The self as a standalone independent core of a being, is unreal. But other things like trees and chairs are not unreal, despite being conditioned, compounded and impermanent.

Nagarjuna taught in his Madhyamaka (Mahayana) philosophy that not just the self, rather, all things are empty of inherent essence. Some people claim this means nothing is real. However this and the concept of papanca in Theravada actually means that things don't exist the way we think they do, not that they don't exist or are unreal.

For e.g. we look at something and objectify it as cooked meat and classify it as delicious food. To a vegan, it may be repulsive. To a honey bee, it's just a lump of dirt because that's not its food.

This means that the cooked meat dish exists but not like how we imagine it to be. How we imagine it to be, is relative to our self. This is how we objectify and classify everything in the world according to our self bias - how is something related to me? It is this inherent essence that is unreal.

Another view from Mahayana Buddhism comes from Thich Nhat Hanh's essay The Fullness of Emptiness. According to this, all things like trees are not independent. They depend on other things. They are made up of things. They are connected to other things. They are constantly changing and not permanent. This is one way to say that it is empty of inherent essence. Not that it's unreal. That means there's no standalone independent object.