Did the Buddha ever refute the view of "extreme metaphysical nihilism" e.g. a view that "nothing is real"?

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.

If we modify your definition of real a teensy teensy tiny bit, then, in fact, the Buddha taught exactly that nothing is real. What's more, this fact is the very key to our salvation. It is precisely because nothing is real that we can awaken and be liberated from the ignorance that anything is real.

For what it's worth, the tiny modification to your definition of real is:

"real: exists and is not like an illusion"

Now, any sharp eyed observer would object with one problem with the above. Emptiness itself, is, in fact, real. That is, emptiness itself exists and it is not like an illusion. It appears exactly as how it actually exists and is non-deceptive in its appearance. This is sometimes called the "Ultimate Truth."

For more, see this answer. And for more evidence see this.

The translation "nihilism" when found in Buddhism merely refers to a denial of or disbelief in the efficacy of kamma, the denial of morality and the denial of causality. The words for this in Pali are natthikavāda, akiriyavāda and ahetukavāda. Refer to MN 60. In other words, there appears to be no term translated as "nihilism" in Buddhism that refers to "nothing is real".

Yes, householder Angus, and interested,

- Namo tassa bhagavato arahato sammā-sambuddhassa -

it's one of the three fatal wrong views:

’When the self that is possessed of form, made of the four great elements,[25] engendered by mother & father, is—with the breakup of the body—annihilated, destroyed, & does not exist after death, it’s to this extent that the self is completely exterminated.’ (DN1)

leading direct to hell and being cause of no grow in Dhamma in this existence. Such views are actually very common under household-worshiping "buddhists", secular approaches, mostly developing equanimity in regard of all mind aspects, yet not aware that being firm attached to form.

See also answer on "What is the practical effect for a Buddhist whose view is materialist?" for more on this.

People with right view regard Paramatta-Dhamma relative, and ultimate in regard of Nibbana, as real.

Citta (mind), cetasika (mind-tendence) and rupa (matter) are sankhara dhammas, conditioned dhammas; they do not arise by themselves, each of them is conditioned by other phenomena.

(Worthy to note here that mind requires matter to a rise and vici-versa, and that mind is the forerunner of phenomena)

In that far: for a person with grave wrong view, it's simply his reality, and so will be the effects from it.

Those Nihilists are so "empty" that they are even not a little nervous that all the bad actions done, holding those views, will fall back to them.

They say repeatedly in this or other ways:

There is no such thing as alms or sacrifice or offering. There is neither fruit nor result of good or evil deeds. A human being is built up of four elements. When he dies the earthly in him returns and relapses to the earth, the fluid to the water, the heat to the fire, the wind to the air, and his faculties pass into space. The four bearers, on the bier as a fifth, take his dead body away; till they reach the burning ground, men utter forth eulogies, but there his bones are bleached, and his offerings end in ashes. It is a doctrine of fools, this talk of gifts. It is an empty lie, mere idle talk, when men say there is profit herein. Fools and wise alike, on the dissolution of the body, are cut off, annihilated, and after death they are not.

Nothing that the Buddha refuted more that the proclaimer of "nothing is real" (worthy to give into) and the generous approach of one of the Buddhas disciples here might help to grasp things not likewise wrong as those lost in their developed world of wrong views, hungry shades in the desert: The Essence of the Dhamma

(Note that this gift of Dhamma is not for trade, exchange, stacks or entertainment but for use in doing merits toward liberation from the wheel here)

He did refute it, but only on the basis of the unskillfulness of following such views. In AN 10:94, Vajjiya Sutta, the Buddha is accused of being a nihilist, to which Vajjiya responds that the Buddha is not a nihilist, as he does not categorically reject nor endorse certain practices, but rather rejects or endorses the skillfulness of them. The Buddha then further clarifies that what matters is whether such views lead to awakening/unbinding, and not the content per se.

Perhaps it's worth complementing the preceding with the Buddha's view on the four kinds of questions. In AN 4:42, Pañha Sutta, the Buddha explains that certain questions are not worth entertaining. It seems to me that matters about the validity of the content of nihilistic views is one of such questions to the Buddha, seeing that he did not spend time responding to the accusations with an argument based on the content per se, but rather on the skillfulness of holding the views themselves.

For even further clarification, in MN 72, Aggi-vacchagotta Sutta, the Buddha responds a series of questions on such absolute views, denying each time that he holds those positions. Even though nihilism is never explicitly addressed, his teaching in that Sutta might help to understand what he could have said on the matter: He explains that positions is something that the Tathāgata has done away with, and that what he teaches is release/unbinding. It seems to me that what he teaches is not that nothing is real or vice versa but, rather, that one can become unbound from this world by the understanding and experiencing of the arising and passing away of phenomena. In my own understanding, it should be noted that for something to pass away, it must have arisen first, so, however empty, this reality "exists" in some way or another.

Abhidharma deals with ultimate realities. Buddha rejected Nihilism (natthikavāda).

The ultimate realities of the world are hidden from us due to our lack of concentration to see clearly and the wisdom to realise things as they are.

Vipassanā helps one see things as they are. Hence there is no view as "nothing is real". For an untrained mind things are not what they seem, but though the 3 fold training you can see things as they are.

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.