Was bronze ever used for chainmail?

"Was it ever?" Certainly. The style of armour evolved with swings of a pendulum as can be seen from the earliest bronze age up until now, with conspicuous heights found in the trench warfare of the First World War.

The Philippino Moro people used bronze and brass in chain mail fashion. The romans used bronze in their loricas. The Philistines had at least one well known warrior:

And there went out a champion out of the camp of the Philistines, named Goliath, of Gath, whose height was six cubits and a span. And he had an helmet of brass upon his head, and he was armed with a coat of mail; and the weight of the coat was five thousand shekels of brass. And he had greaves of brass upon his legs, and a target of brass between his shoulders. (1 Sam 17: 4–6, KJV.)
_{Although to be fair, "mail" would have been an appropriate translation of the Hebrew in most times gone, "chain mail" is now anachronistic and the whole passage probably means flexible snake-like segment mail. The ESV uses just "mail". Vulgate used "et cassis aerea super caput eius et lorica hamata induebatur porro pondus loricae eius quinque milia siclorum aeris"}

The main reason for adopting rusting and brittle iron if you had good bronze was that you sometimes just didn't have the raw materials for bronze, at all. This availability encompasses the problem of price for imported goods (Cf. ch Why Change from Good Bronze to Bad Iron?) Early iron was of horrible quality and needed a significantly altered process structure and new knowledge and experience to work with. While later steel was in most military aspects superior to bronze, reaching that level took quite a while.

Re-enacters today face a similar choice in which metal to use:

Brass, Copper, and Bronze: These metals give a penny like appearance and are rather strong. They are all expensive; however, copper is especially expensive. They are all very corrosion resistant, making them another good choice for Functional Armor. These metals are also quite heavy. (Just like steel, the comparison was with aluminium)

Bronze being an alloy has a broad range a different formulations. Decrease or increase tin and you control a large part of hardness or brittleness, annealing capabilities and so forth. This requires considerable skill, but the bronze age lasted for quite a while so that the early metal workers could really master their trade. And this process of possible refinement continued alongside throughout the time after iron working was introduced. One important advantage that iron had over bronze was that the Romans quickly learned how to recycle used iron, often increasing its quality, whereas they mostly tried just the same with bronze and degraded its quality in the process.

How effective is it?

Generally speaking, mail's resistance to weapons is determined by four factors: linkage type (riveted, butted, or welded), material used (iron versus bronze or steel), weave density (a tighter weave needs a thinner weapon to surpass), and ring thickness (generally ranging from 18 to 14 gauge (1.02–1.63 mm diameter) wire in most examples).

You do not need iron to make mail, but you need the idea, the pattern and the tools to make it. It is not entirely clear who invented it, tough.

John W. Eadie: "The Development of Roman Mailed Cavalry", The Journal of Roman Studies, Vol. 57, No. 1/2 (1967), pp. 161-173:
The rider is almost completely encased in bronze or iron: a one-piece mask-helmet covers his head entirely with the exception of eye-slits; his body, from shoulders to knees, is protected by a mail suit constructed of small, overlapping bronze or iron plates, which are sufficiently pliant to permit freedom of movement; attached to the mail coat are greaves to protect his legs and feet. The horse is similarly armoured: head covered by a metal plate; back and flanks protected by a 'blanket' of thin iron plates; legs fitted with metal greaves. Only the horse's belly and presumably his eyes and tail are unprotected.

From the British Isles, Denmark to the Parthians at least are there examples over space and time it seems.

And it is also not necessarily a purity concept to use only one type of metal for such an outfit. Iron, steel and bronze are fundamentally different, but each of them can be made in a staggering variety of qualities and attributes. So, using the right tool for the right job, things were sometimes combined.

E. M. Jope: "A Fragment of Chain-Mail from the Romano-British Temple Site at Woodeaton", Oxoniensia, Vol. XXII (1957) pp. 106-107:
Such chain fabric has been found in pre-Roman and Roman contexts on the continent and in Britain. It is a remarkable technical achievement, especially the fine riveting.5 In some cases there is no doubt that it was chain-mail,6 and in most others it probably was.7 The bronze links in this Woodeaton piece are simple rings of thin bronze rod 1 to 1.1. mm. thick, bent round so that the ends are butted to form a circle 7.2 to 7.5 mm. across. As with the iron links, this again is a widely used standard ring size and rod gauge, within narrow limits. Similar bronze links have been found attached to iron chain fabric as bordering or for attachment of ornaments. Many of the objects from the temple area at Woodeaton were evidently votive offerings, and as pieces of chain-mail had been deposited in the bog at Thorsbjerg (Denmark), some carefully placed in pottery vessels, this piece might similarly have been a votive offering at Woodeaton. However, parts of bronze chain regalia, made up of butted bronze links just like those attached to the Woodeaton piece, have been found at other Romano-Celtic temple sites, and it is possible that this iron chain fabric decorated with bronze links might have been similarly used. though it could have been merely lost, it is perhaps most likely that it was a votive offering.

Some may object that bronze is unsuitable in general and impossible to work with in this kind of application. It seems at the least the Romans would object to that. (Admittedly, iron was used far more often.)

David Sim: "Roman Chain-Mail: Experiments to Reproduce the Techniques of Manufacture", Britannia, Vol. 28 (1997), pp. 359-371:
According to Singer draw-plates for drawing wire were unknown until the tenth century A.D. and wire was first produced by forging. However Thomsen and Thomsen have stated that non-ferrous wire from Persia of sixth- to fifth-century B.C. date could only have been made by wire drawing. Northover has argued that two bronze plates found at Isleham, Cambridgeshire, thought to belong to the Late Bronze Age, are draw-plates. These draw-plates are assumed to be for drawing non-ferrous metal. The drawing of iron wire is more difficult because the higher tensile strength of non-ferrous metals requires a much greater force to bring about drawing.[…]
Two iron plates from the Roman period have been shown to exhibit all the characteristics of draw-plates, but it cannot be proved that they were used to draw iron wire. However, experiments have shown that iron wire can be drawn through iron draw-plates and that such draw-plates are simple to produce. During experiments the force required to draw iron wire was between 1415 N and 2094 N. Exerting this force by hand reaches the limits of human strength, but a simple system of pulleys would have been able to produce the necessary force. The dimensional accuracy of the wire used to make Roman riveted rings was to a level that the author could not reproduce by forging and filing or by swaging. It is suggested that the accuracy of the rings and the time to produce the necessary quantities of iron wire for riveted rings means that the wire was probably made by drawing it through draw-plates. The apparent scarcity of equipment for drawing iron wire can perhaps be attributed to some draw-plates having been misidentified as nail-heading tools, as well as the fragile nature of iron items in the archaeological record.

Some finds are much more durable, though. And the best thing about bronze is that it doesn't rust so much.

This tiny piece of bronze mail was found at Lunt Fort, Bagington, Warwickshire, constructed during the revolt of Boudicca, and abandoned soon after around AD 80 (photo). It was in use again in the 3rd century.

The Netherlands have yielded both examples of lorica hamata squamataque; see the article by Martijn A. Wijnhoven above. (Source: The Evidence for Roman Chain Mail)

If iron would be so superior it would have all but supplanted the use of bronze, once it was a mastered technology? Not for the Romans, at least.

Michael Fulford, David Sim and Alistair Doig: "The production of Roman ferrous armour: a metallographic survey of material from Britain, Denmark and Germany, and its implications", Journal of Roman Archaeology, Volume 17 2004 , pp. 197-220:
Even though steel armour was more protective than bronze and, if polished, might have been thought more visually impressive, the sheer quantity of what was needed to equip the army probably explains why bronze and ferrous armours continued side by side to the late empire.

Similar mixtures of metals in various applications can be found in use by cataphracts:

Cataphracts were almost universally clad in some form of scale armor (Greek: φαλιδωτός Falidotos, equivalent to the Roman Lorica squamata) that was flexible enough to give the rider and horse a good degree of motion, but strong enough to resist the immense impact of a thunderous charge into infantry formations. Scale armor was made from overlapping, rounded plates of bronze or iron (varying in thickness from four to six millimeters), which had two or four holes drilled into the sides, to be threaded with a bronze wire that was then sewn onto an undergarment of leather or animal hide, worn by the horse. A full set of cataphract armor consisted of approximately 1,300 or so "scales" and could weigh an astonishing 40 kilograms or 88 pounds (not inclusive of the rider's body weight). Less commonly, plated mail or lamellar armor (which is similar in appearance but divergent in design, as it has no backing) was substituted for scale armor, while for the most part the rider wore chain mail.

There is a certain notion of inherent progressivism in history, limited understanding or premature conclusionism of or in material science that seem to be in need of addressing them. Iron is not simply "better" than bronze. In fact, often bronze is the better material for the job at hand. "Iron" is just as precise a description as "carbon" is when you want to describe the absolutely obvious inherent hardness of diamonds; or graphite:

It would be made by almost anyone who is unschooled in materials science, and by a good many who claim intimate knowledge of the subject. To the average person the properties of a material are uniquely associated with its name, which is usually derived from its dominant chemical constituent or the whim of the company that manufactures it. Almost everyone knows that the Iron Age succeeded the Bronze Age. The idea that "iron" might be inferior to "bronze" is a possibility that a person who does not know metallurgy is unlikely to consider. Even physical scientists of impeccable credentials often assume that the properties of a material are uniquely associated with the atoms that make it up, and that those properties would be thoroughly understood if we only mastered behavior at the atomic level.
This notion is wrong. The material we call iron can be made weak (easily bent) or strong (virtually impossible for a human to bend), ductile (capable of being bent or deformed into complex shapes without fracture) or brittle (easily broken). Examples of all of these manifestations of iron are common today and useful in engineering. It follows that when we describe a material as iron we have left out something important. In fact, we have left out something essential.
That thing that is missing from the designation, iron, is called microstructure. While the term "iron" describes the nature of the atoms that are present, or at least the dominant atom type, the term "microstructure" describes how those atoms are arranged. Both are necessary to understand the properties of iron. The same is true of any other engineering material. The composition and the microstructure together define the material; they specify what it is and what engineering properties it will have. Either, alone, is insufficient.[…]
Early iron was inferior to good bronze, but it wasn't that bad. And it was plentiful and cheap. Given a choice between a thousand soldiers armed with iron and half that number armed with bronze, the wise king invested in iron. In many societies of the period soldiers were expected to provide their own weaponry. Given that he could afford fifty arrows tipped with iron or twenty tipped with bronze, the smart soldier made up his mind very quickly.
(From: Morris/Survey, above.)

This would all be debatable. It's just archaeology and therefore a bunch of rusty metals in a heap of dirt. And after all this evidence there might be still some doubters whether bronze can be made into wire? Modern minds need modern evidence.

It's even used today by some fashionable pepople: All categories “bronze chainmail” (1,018 Results)

Bronze went out of fashion, but it never went out of business.

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Just out of a curiosity:
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