I had a thought, are there any conceivable fighting styles that would generate a focus on armoring the limbs, like gauntlets on the arms and boots on the legs, over the main body? Does anyone have any thoughts on how such a style might work?
I highly doubt that there is an existing fighting style that prioritized protecting extremities over protecting the body for fairly obvious reasons.
For such a fighting style to be viable or practical there would need to be a complete a utter lack of ranged attacking, no bows, no slings, no throwing weapons. The existence of any ranged attacking imposes far too much of a risk to the core for it to be left unprotected. The existence of grappling or wrestling techniques would similarly impose too great a risk for this style to be viable, as soon as the fighter is not able to freely and effectively move their extremities they are open to lethal damage.
With that in mind, realistic armor that protects the vitals would also make this fighting style a very poor choice. An opponent who largely risks superficial damage to their limbs vs one who risks lethal damage to their vitals at a single mistake puts the odds heavily in favor of protecting vitals.
That restriction in place, without protecting the body you would want fighters to move quickly to cover their blatant openings and to present a threat to their opponent as often as possible to try and keep distance and avoid the above mentioned risk of grappling.
Overall I would say that unless this is meant to be fated as some sort of spectacle fighting, such as capoeira, it would never exist in real combat.
There are lots of practical problems with this sort of armor, as BKlassen addressed. But such a fighting style is introduced in Slaves of the Mastery, the second book of the The Wind on Fire trilogy, and used briefly in the third.
In the book, the style (called the Manaxa, which also refers to the gladiatorial event) is used by pseudo-gladiators who only fight other practitioners. Gladiators wore a bladed gauntlet on each hand, a spiked helmet, and bladed shinguards, all of which were used offensively but granted a modicum of protection, provided the fighter was quick enough.
While I was unable to find a good text copy for an excerpt, as AlexP noted it tended to work for only a very short time. Speed was paramount; fights were short, bloody and one-on-one, and occasionally ended in death.