Underestimate not the ingenuity of people before refrigeration. Especially not the peoples of what we now call the Middle East. More can always be said if anyone would like to speak about pre-refrigeration ice mining or ice making, especially since this is an awesome subject that can always use more posts!

For the meantime, OP, grab your gloves and put the Frozen on, because we're going ice mining.

• u/Bentresh looks at how they did it in the Bronze Age Near East;
• u/J-Force looks at how they did it in the Medieval Near East;
• and u/Osarnachthis points out that the Ancient Egyptians had a word for snow.

Bonus: Bentresh mentions them in their answer, but for completeness, here is the Encyclopedia Iranica article on the Iranian yaḵčāl.