Sumer, sometime in the 4th or 3rd millennium B.C.

My first instinct was to check out the Wikipedia article again. One interesting quote was

The presence of dragons within Chinese culture dates back several thousands of years with the discovery of a dragon statue dating back to the fifth millennium BC from the Yangshao culture in Henan in 1987, and jade badges of rank in coiled form have been excavated from the Hongshan culture circa 4700-2900 BC.

That would seem like a likely candidate. Unfortunately, the in-text citations do not work, I can find no other mention of either of these that does not quote Wikipedia verbatim, and this probably doesn't count as mythology, either. Neither are associated with "myths".

I stumbled across this blog post/whatever-it-is. Either way, it was helpful insofar as it led me somewhere. Citing Sumerian Mythology, 1944, 1961, by; Samuel Noah Kramer, it states

KUR Sumeria 3rd Millennium B.C. "Since the dragon-slaying theme was an important motif in the Sumerian mythology of the third millennium B. C., it is not unreasonable to assume that many a thread in the texture of the Greek and early Christian dragon tales winds back to Sumerian sources." . . . We find mention of Kur in three myths from the 4th - 3rd Millennium B.C., (more than a millennium before Tiamat!), In the introductory prologue to the epic tale "Gilgamesh, Enkidu, and the Nether World,"( written on eight tablets - seven excavated in Nippur and one in Ur), Where Enki, the water-god, fights Kur after he learns that The goddess Ereshkigal was carried off violently into the nether world, by Kur. Enki fought Kur from a boat, and Kur fought back savagely with stones of all sizes, and attacked Enki's boat with the primeval waters which it controlled. Unfortunately for us, the author of this tale is so anxious to proceed with the Gilgamesh tale that he doesn't finish the dragon part, and leaves us hanging. It is certain that Enki wins though because he is in the rest of the poem, Kur is not. . . . The second version of the slaying-of-the-dragon myth can be found in "The Feats and Exploits of Ninurta." (49 tablets) A significant version, due to the fact that it is evident that it was utilized by the Semitic redactors in the creation of the Babylonian Creation Myth featuring Tiamat. In this version, Ninurta, the warrior-god, is the hero of the story. His personified weapon, Sharur, kisses up to him in a drawn out speech extolling the heroic qualities and deeds of Ninurta to convince him to go after Kur, and attach and destroy him. What Sharur has against Kur is not written in the text that is available. Ninurta leaves to do as asked, but finds himself lacking and "flees like a bird". Sharur though, won't let it go and speaks, reassuring and encouraging Ninurta with his words. "Ninurta now attacks Kur fiercely with all the weapons at his command, and Kur is completely destroyed." . . . The third version of the slaying-of-the-dragon myth can be found in "Inanna and Ebih." A one hundred and ninety line poem. (12 tablets) The dragon-slayer in this version of the story is a goddess, Inanna, curiously known as both the goddess of love and also as the goddess of battle and strife, (She must have been married), and is also referred to in many Sumerian hymns as "The Destroyer of Kur." Kur, is also referred to as The 'mountain,' in the Poem. Did I mention that Kur was also the first fire breathing dragon?

Interesting. I decided to learn more.

First, the relevant part of Sumerian Mythology appears to be available here. No part of it quotes any relevant sections of the myths that imply or outright state that Kur is a dragon. So I decided to find them for myself. (By the way, related notes on many Sumerian myths, including these, can be found here).

Gilgamesh, Enkidu, and the Nether World

The relevant passage appears to be

1 It began in the early days of the world, after the earth and the heavens had been separated. It was a turbulent time. Ereshkigal had been taken by the dragon Kur and Enki was sailing after to put things into the order that they belonged in. 2 The waters that Enki sailed were being pummeled by rocks. The smallest of these hit the side of Enki's boat like they were like hammers. The largest of these hit with the impact of a boulder thrown from a catapult.

This is all that remains of the explicit tale of the fight of Kur against Enki.

The Exploits of Ninurta

The Asag leapt up at the head of the battle. For a club it uprooted the sky, took it in its hand; like a snake it slid its head along the ground. It was a mad dog attacking to kill the helpless, dripping with sweat on its flanks. Like a wall collapsing, the Asag fell on Ninurta the son of Enlil. Like an accursed storm, it howled in a raucous voice; like a gigantic snake, it roared at the Land. It dried up the waters of the Mountains, dragged away the tamarisks, tore the flesh of the Earth and covered her with painful wounds. It set fire to the reed-beds, bathed the sky in blood, turned it inside out; it dispersed the people there. At that moment, on that day, the fields became black potash, across the whole extent of the horizon, reddish like purple dye -- truly it was so!

Asag is often considered to be different from Kur. However, this (of dubious reputation) states

Are Asag and Kur synonymous, both referencing a dragon? Or are they different entities attributed as the dark force overcome by Ninurta? Owing to the age of Sumerian literature, and its predecessors adapting the myths, this remains uncertain.

Inana and Eibh

An, the king of the deities, answered her: "My little one demands the destruction of this mountain -- what is she taking on? Inana demands the destruction of this mountain -- what is she taking on? She demands the destruction of this mountain -- what is she taking on? "It has poured fearsome terror on the abodes of the gods. It has spread fear among the holy dwellings of the Anuna deities. It has poured its terror and ferocity over this land. It has poured the mountain range's radiance and fear over all the lands. Its arrogance extends grandly to the centre of heaven. . . ." My lady confronted the mountain range. She advanced step by step. She sharpened both edges of her dagger. She grabbed Ebih's neck as if ripping up esparto grass. She pressed the dagger's teeth into its interior. She roared like thunder. The rocks forming the body of Ebih clattered down its flanks. From its sides and crevices great serpents spat venom. She damned its forests and cursed its trees. She killed its oak trees with drought. She poured fire on its flanks and made its smoke dense. The goddess established authority over the mountain. Holy Inana did as she wished.

Huh. That's disappointing. The literal interpretation states that Inana's foe is a mountain, not a dragon. Indeed, as that earlier dubious source states,

Another disambiguation should be made between Kur the dragon and the myth of Inanna and Ebih. Inanna was the Sumerian goddess of love, battle, and strife and among her many epithets was the 'destroyer of Kur.' In this myth, Kur is also called 'mountain Ebih,' which is a district northeast of Sumer. Therefore, Inanna overcame Kur, but in this myth, Kur is an hostile land, not a dragon.

Is the myth being metaphorical when it describes Inana's foe as a mountain?

This myth is doubtful, as are, come to think of it, the other two.

In my (non-expert) opinion, Gilgamesh, Enkidu, and the Nether World is not strong evidence for a dragon. The source I gave is the only translation I could find that explicitly calls Kur a dragon. Most others do not refer to that. Inana and Eibh also seems to be a lost cause, because, quite literally, Inana is fighting a mountain, not a dragon. I doubt that the myth is being figurative. Mythology tends to be literal, realistic or not.

The Exploits of Ninurta seems to be the best evidence available. It says

like a snake it slid its head along the ground. . . . like a gigantic snake, it roared at the Land. . . . It set fire to the reed-beds

Circumstantial? Perhaps. But it seems like a dragon-like creature. I'm calling this one the winner.

Now for the date. Embarrassingly enough, I haven't found any date for this other than the one Kramer gave, somewhere in the 4th or 3rd millennium B.C.