In most cases, the answer would be a rounding error. In especially lopsided battles, in which great numbers of the enemy were massacred without being able to fight back, the average may have approached 1.

We don't have many accounts of individual feats of arms in the battles of antiquity, but we can get a ballpark figure by looking at overall numbers. I've written before about the casualty rates in ancient Greek battles. As it says there, the losses on the losing side (on the basis of a very imperfect data set) seem to have averaged about 14%. Assuming a rough parity of numbers, that average means the vast majority of men in a victorious army never killed anyone. At most, assuming an even spread, about 14% of them killed one man, while the other 86% killed zero. The average number of men killed would therefore be 0.14.

If this is too abstract, we can look at a concrete example. The Athenians fought a battle against the Boiotians at Delion in 424 BC and suffered a catastrophic defeat. The Boiotian army of 7,000 hoplites, 1,000 cavalry, 500 hamippoi and about 10,000 other light-armed troops inflicted about 1,000 casualties (plus an unknown number of dead light troops and camp followers). Most of these seem to have been caused by the cavalry during the chaotic Athenian flight, but we don't have the details to be precise about this. Assuming an even split, there is one dead Athenian for every 18.5 Boiotian combatants. The average number of men killed by each warrior was 0.05.

These are armies using similar equipment and tactics; as Peter Krentz noted in the article cited in the linked post, casualties tended to be more extreme in encounters between different tactical systems (though our sources are also less reliable). The Athenians lost 1,000 men at Delion, which was a tragedy; but they celebrated inflicting as many as 6,400 losses on the Persians at Marathon. If we accept the later evidence that their own army consisted of about 9,000 Athenian and 1,000 Plataian hoplites, we can see a very much higher average. In this historic victory, the average defender may have killed 0.64 Persian troops.

We hear of much more extreme figures for the battle of Thermopylai - 4,000 Greek dead against 20,000 Persians - but the Greek tally is much more certain than the Persian. The number of 20,000 dead comes from a story that Xerxes was embarrassed by his losses, quickly hid 19,000 of them, and displayed the remaining 1,000 to his army and fleet alongside the 4,000 Greek dead. In other words, our source for this story (Herodotos 8.24-25) only talked to people who saw 1,000 Persian dead on display. The other 19,000 are probably an embellishment. If we decide to believe his story, the average Greek at Thermopylai may have killed an astonishing 3 or 4 Persians (depending on how many Greeks there were in total), but if we choose to be more skeptical, the average is in the more realistic 0.14-0.19 range.

It is easy to understand why the average was generally well below 1 when we consider how ancient armies fought. Heavy infantry like the Greek hoplites were drawn up in close-order formations many ranks deep. Battles would not go on until these formations were ground to a pulp, but would be decided when the morale of one of the two sides collapsed and the troops fled. By that point, the vast majority of the troops in the formation would not actually have done any fighting. They were there to give solidity to the formation, to help it stand its ground against enemy charges, and to keep the men in front from running away. But the actual fighting was done by the men in the first two or three ranks. In the Hellenistic period, when pike formations might be as many as 16 or 32 deep, the men in the leading ranks were deliberately chosen for their size and ferocity, to ensure that the hard work would be done by those most suited for it.

So can we break down the total number of casualties by group? Can we select for those men who actually fought, in order to get a more accurate sense of how many men they might have killed? Most of the time, this is impossible; the sources do not break down the numbers in any detail. In any case, they often stress that the lion's share of the killing wasn't done by heavy infantry fighting in the ranks, but by the light infantry and cavalry pursuing a fleeing enemy. It is unlikely that any Spartan hoplite ever killed as many men as the Celtic mercenary cavalry in Spartan service at the Tearless Battle in 368 BC, since the horsemen's enemies weren't fighting back. But we can use the number of total casualties to get something of an estimate. The losing side would take most of its losses in the rout; but the winning side also incurred casualties, to an average of about 5%. These men must have died in the battle to overcome the enemy front line. If we assume no further losses were suffered after the enemy broke, this number gives us a better indication of how many men were killed by the front rank(s) of a hoplite formation.

Take the battle of Delion again. We know there were about as many Athenian hoplites as Boiotians, and they were drawn up 8 ranks deep. That gives a front rank of 875 hoplites. We are told they inflicted just shy of 500 losses on the enemy before the Athenian army broke. Here, then, the average men killed per hoplite is much higher - about 0.57. But we are also told that Boiotian casualties were especially high among the men of Thespiai (one of the cities in the region), who were surrounded by the Athenian right wing and fought to the death. We can assume that the majority of the Boiotian dead belonged to this contingent. Here, on the narrow part of the line that faced the Thespians, and in the unusual circumstance of an encirclement, single Athenian hoplites may have killed several men.

Something similar may be said about the battle of the Nemea in 394 BC. Here, a Spartan-led alliance collapsed on contact with a vast anti-Spartan coalition, but the 6,000-strong Spartan contingent kept its nerve and was able to roll up the entire enemy line. We are told that the anti-Spartan coalition suffered as many as 2,800 casualties. Spread over the whole Spartan army, this would yield an average of perhaps 0.19, if not less; but the actual killing was overwhelmingly done by the Spartans themselves, whose average murder rate may therefore have been about 0.5. Unfortunately we do not know the depth of the Spartan formation, so we can't break this down further by rank, but it is quite likely that the men in the front ranks on this occasion, attacking unit after unit of confused and panicked enemies, may have achieved slaughter at a rate of 3-4 kills per man.

Statistically, though - as I've explained in previous answers - it would be unusual for a participant in an ancient battle to kill a man, and extremely rare indeed for a heavy infantryman to kill several.