Did The Romans Ever Have Iced Drinks?

Yes. It seems they might have been as fond of putting ice in their drinks as we are today. It wasn't, however, quite as convenient as going to the freezer compartment of the refrigerator and taking out a tray of ice cubes. They had no way of making ice, so it all had to be transported by mule from the mountains.
It became one of the ways of showing wealth at the great parties held in Roman times, to allow guests to freely sprinkle the refreshing ice chunks in their drinks. It was denounced by some as a sign of the growing decadence. Seneca who was tutor to the young Nero. He wrote "You see skinny youths wrapped in cloaks and mufflers, pale and sickly, not only sipping the snow but actually eating it and tossing it into their glasses lest they become warm simply through the time taken in drinking."