This question gets asked here from time to time, which is why there is a section of the FAQ dedicated to it.
It is also a controversial topic among scholars. On the one hand, you have authors like Jonathan Shay and Lawrence Tritle, who have argued that there are clear cases of PTSD visible in our ancient sources, showing that psychological and physical problems caused by combat stress are a universal part of the human experience. On the other hand, you have authors like Jason Crowley, who argue that the nature of warfare and its place in society were fundamentally different in ancient times, and that the kind of trauma caused by modern, industrialised warfare - massively lethal and completely divorced from civilian life - would have been unknown to the ancients.
The first school of thought has a couple of favourite pieces of evidence, from Classical Greek sources in particular. The most famous by far is Herodotos' anecdote about the fate of an Athenian named Epizelos, who fought the Persians at Marathon in 490 BC.
An Athenian, Epizelos son of Kouphagoras, was fighting as a brave man in the battle when he was deprived of his sight, though struck or hit nowhere on his body, and from that time on he spent the rest of his life in blindness. I have heard that he tells this story about his misfortune: he saw opposing him a tall hoplite, whose beard overshadowed his shield, but the phantom passed him by and killed the man next to him.
-- Herodotos 6.117.2-3
In addition to this, some have seen PTSD lurking behind the character of a Spartan commander named Klearchos, who was said to be violent and quick to anger, and whose long exposure to warfare was supposed to have made him unsuited for any other environment:
Such conduct as this, in my opinion, reveals a man fond of war. When he may enjoy peace without dishonour or harm, he chooses war; when he may live in idleness, he prefers toil, provided it be the toil of war; when he may keep his money without risk, he elects to diminish it by carrying on war. As for Klearchos, just as one spends upon a loved one or upon any other pleasure, so he wanted to spend upon war— such a lover he was of war.
-- Xenophon, Anabasis 2.6.6-7
Others have also looked for PTSD in the tragedy Aias by Sophokles. In this play, the hero Aias is driven to fury after he loses to Odysseus in the competition for the dead Achilles' armour. He plots to kill Odysseus and the other leading Greeks, but Athena fools him into killing a herd of cattle instead. When the red mist subsides and Aias realises what he has done, he is so overwhelmed with shame that he commits suicide. These are the 3 main examples of PTSD-like symptoms found in the Greek sources.
The counter-argument raised by those who argue against PTSD's universality is that none of these examples offers anywhere near enough information for us to make a confident diagnosis. It is generally bad medical practice to make a diagnosis without seeing or interviewing a patient, which is of course impossible in the case of historical figures. Strictly technically speaking, no person could be diagnosed with PTSD before that disorder was officially defined in the 1970s. In cases where we have plenty of evidence from eyewitnesses and doctors about a person's condition, we are of course on firmer ground, even if the case dates before the 1970s. But the above examples obviously do not give anywhere near enough detail for a diagnosis to be made.
The same may be said of a number of Assyrian medical texts that have recently been cited as evidence of ancient PTSD. These texts effectively describe little more than sleepwalking and nightmares, just as Herodotos' anecdote describes nothing more than hysterical blindness, and Klearchos' biography only a mind hardened to military life. You can find more detail in this post by u/kookingpot.
In this sense, the answer to your question is "no" - not because people weren't suffering psychological trauma (they very probably were), but because our sources are nothin like modern psychological evaluations, and we cannot diagnose a dead person with PTSD (or any other form of mental illness).
Scholars like Crowley go further than this, arguing that the disorder and symptoms we describe as PTSD are a result of the cultural and technological conditions of modern warfare, and that they essentially didn't exist in the ancient world. They point to the ubiquity of violence and death in the civilian life of the ancient Greeks, and the ubiquity of war and martial values in their society in general; they point to the fact that citizen levies went to war side by side with their family and friends, who could support them both at home and on campaign; they also point out that elaborate rituals existed, at least in Classical Athens, to facilitate the warrior's return to his normal life and environment. As a result, they argue, war as an experience was far less traumatic and alienating. It's only in exceptional cases that anyone might suffer lasting consequences from traumatic experiences, like Epizelos did. It is for this reason that we never hear of ancient sources discussing combat trauma as a common occurrence among warriors, and that the ancients never discuss ways to treat anything like PTSD or its symptoms.
The reason why this counter-argument is intensely controversial is that it runs afoul of real people with real problems who are looking to scholarship for answers. Thanks to the works of Shay and Tritle, however scientifically questionable their conclusions, the ancient source material is now being used to help people come to terms with combat trauma (and other forms of PTSD). Various projects exist that use Sophokles' Aias to get veterans to talk about and process their experiences. This sort of thing is meaningful and helpful, even if it plays fast and loose with our sources to do so.
Within the context of these efforts, not just to help individuals, but also to remove the stigma of PTSD as a form of "cowardice" or "weakness" by establishing its universality, it is regarded as harmful and crass to point out that perhaps we really don't have any basis to suppose the ancients suffered PTSD. Our sources do not give us the evidence, and the ethical standards of medicine prohibit facile generalisations.
One way forward that is currently being explored starts from the premise that, just because the ancients didn't have a word for it and never described it, that doesn't mean they weren't suffering similar forms of trauma to those exposed to extreme conditions today. Using evidence from neuroscience, we can track the physical changes in the brain that occur among those traumatised by combat, which in turn cause the symptoms of PTSD. These changes are a neurological reaction to an overload of the fight-or-flight instinct, and this seems to be hardwired into our nature as human beings. If we may assume that the human brain has not changed its physiology in any fundamental way over the last few thousand years, then the ancients, too, must have suffered what we call PTSD, even if they may never have thought to classify it as a disorder, understand its causes, or try to find a way to treat it.
Reading
J. Shay, Achilles in Vietnam (1995)
L.A. Tritle, From Melos to My Lai (2000)
J. Crowley, 'Beyond the Universal Soldier: combat trauma in Classical Antiquity', in D. Konstan/P. Meineck (eds.) Combat Trauma and the Ancient Greeks (2014)
PTSD in Ancient History, or how to disagree amicably by Owen Rees (2014)
As a follow up question I would be interested to know how ptsd would have been treated?