Has anything changed in Ukraine recently? Is ТриМоб still the only 3G network in Kiev? Is there any 4G yet?

Is it possible to buy a ТриМоб nano-SIM at Kiev Бориспіль airport (ideally in Terminal D)?

Unfortunately nothing has changed. ТриМоб is still the only 3G network in Ukraine (if we doesn't count CDMA networks). 3mob has the worst marketing and PR departments in the world I guess. Half of friends of mine still thought that there're no any 3G options except CDMA.
You can get micro or nano sim in mobile phone shops and also in the 3mob offices. I don't know about situation in the KBP.

Thanks very much for the advice. Does ТриМоб definitely issue prepaid SIM cards? On their web site , it suggests it's contract only.

I've been in Kiev since Friday. I tried to buy a ТриМоб SIM card in the centre of Kiev but only an independent shop would sell me one with a huge markup, bringing the price to around UAH 130. As a comparison, the same shop had a price for a Kyivstar SIM of around the same amount. I then went to a Kyivstar shop and bought a new SIM for UAH 10 (including UAH 5 of credit I think), which I then swapped to a genuine nano-SIM for UAH 21. Although this only gives me 2G, it's fine for background data and free wifi is ubiquitous in Kiev. Often the first thing people do when they sit down in a bar or restaurant is ask for the establishment's wifi password.

Unlike ТриМоб, I can keep my Kyivstar nano-SIM active for future trips with only a balance deduction every 12 months. The helpful girl in the Kyivstar shop told me I needed to switch off data (UAH1/day) by dialling *154*0# before leaving Ukraine and reactivating it on future trips by dialling *154*1#. If you don't switch it off, you are charged UAH1/day even if you don't use data on a particular day.