Although it sounds unbelievable, in ancient times (even in the desert) they had the ability to store ice.
There are a few different sources for this information online. Firstly, there is this article written by some Malaysian architecture students. Their research stated:
In 440 BC Iran, Iranian people had already domination the technique
of storing ice in the middle of summer in the desert. The ice was
brought in during the winters from nearby mountains in bulk amounts,
and stored in a specially designed, passively cooled refrigerator,
called an Earth refrigerator. This was a large underground space that
had thick walls made out of a special mortar called sarooj, composed
of sand, clay, egg whites, lime, goat hair, and ash in specific
proportions, and which was resistant to heat transfer.
This is backed up by other sources, such as the medical book Cutaneous Cryosurgery, which look at this issue in relation to ancient medicine. It stated:
At first glance it is not clear how ice or snow would have been
available at such latitudes, but even in hot countries there were
means of acquiring it. Ice could be stored, from the winter time, in
ice houses where it was packed in large quantities and covered with
straw or other insulating materials...Ingenuity was at its foremost
when methods were developed to produce ice or slush in desert areas
when extreme low temperatures at night were manipulated to freeze
evaporating water. The Romans and later Iranians would dig a pit and
line it with insulating straw. In it was placed a water contained and
the opening was covered by a sun-reflecting shiny metal in the day but
open to the elements at night. Evaporation at night led to ice forming
around the edge of the container. This was collected and stored.
So as unusual as it might as first sound, the ancients certainly had the ability and know-how to do this.