So, how you answer this question depends how you define a 'handgun'. As a modern-day colloquialism, 'handgun' is an equivalent term to 'pistol', i.e. a firearm small enough to be comfortably wielded with just one hand. However, historically, the term 'handgun' emerges out of the 'hand-gonne', which referred to any firearm portable enough to be carried in both hands and operated by a single person. These typically took the form of a very small cannon affixed to the end of a small pole for stability. And in that sense, it is pretty uncontroversial to suggest that the first firearms were handguns – using that more archaic definition. I've discussed the relevant firearms history in answers such as this one, this one, and this one.

To try to give a more specific accounting of things in this context, gunpowder's early military uses were as a relatively stable incendiary, rather than as a propellant. The recognition that certain mixtures might have greater explosive potential was recognised by Chinese weaponmakers, who developed small hand grenades and, eventually, the 'fire-lance', first definitively employed in the 1130s but possibly being in limited use as early as 1000. The fire-lance was a single-use device that had a gunpowder charge housed in a typically bamboo tube which was used to project a plume of flame outwards, with variations including metal, stone, or ceramic fragments as well. The general suggestion is that early hand-guns, and in turn larger anti-personnel-calibre cannons, emerged out of making reusable fire-lances with metal chambers instead of bamboo or other organic materials. So depending on whether you count the single-use 'fire-lance' or not, we can definitively date the medieval hand-gun to either the 1130s (when the bamboo examples are definitively attested) or the 1290s (when we have the earliest definitive example of a metal gun from western China).