A hovering weapon the size of a hand phaser would seem to be beyond TOS technology.
What about Flint's M-4?
TOS didn't dwell on hoverbots, for obvious reasons, but it's not like that couldn't exist.
We know from hearsay in "The Immunity Syndrome" that Federation starships can launch probes into hazardous areas so that the mothership can hang back in relative safety while gathering intelligence. We know a hand phaser, small enough to fit in your palm, is powerful enough to disintegrate a creature the size of one of Neural's Mugatos. So it's not a stretch to consider the possibility that armed probots could exist. Maybe the whole thing wouldn't be the size of a phaser, but that would probably be because, like TOS communicators, tricorders, and other gadgets, these would have to be seen as field equipment items, serviceable by "stone knives and bear skins".
I assume they would be frowned upon by the Federation. Armed, unmanned probots "can't think", in other words, they can't make judgement calls that a human field officer should place his/her faith in. In an Universe where Federation explorers are bound by rules like the Prime Directive, and no doubt concerns about probes falling into the wrong hands, their use may be severely limited. We pat ourselves on the back for finding clever ways to wage war with remote-controlled drones armed with Hellfire missiles today, but if we had to fight a more formidable foe that could jam our communications or intercept our machines, we might discover how fragile and potentially dangerous that kind of strategy could be. (That, and it should be noted that it was "boots on the ground" that went after bin Laden, not a drone.)
So, on the one hand, I would not be surprised if Kirk's Enterprise had at least some limited storage of combat-capable probot/hoverbot machines tucked away. They would've come in handy on Cestus III, but that's the obvious nature of interstellar confrontations in TOS: "they're never there when ya need 'em".