Yes, they do exist within the game coding (and like previous games, are reserved for event Pokémon only).
Infact, the Shiny Pikachu/ Shiny Eevee(Depending on your game) in a event from a while ago comes in a cherish ball.
I even found serebii and bulbapedia sources to confirm this.
https://m.bulbapedia.bulbagarden.net/wiki/List_of_Korean_region_serial_code_event_Pokémon_distributions_in_Generation_VII#Pikachu Scroll down to Shiny Pikachu. From game “E” Eevee.
https://www.serebii.net/events/dex/025.shtml Look at the most recent event Shiny Pikachu from LGPE. On its discription it shows what ball it comes in on the upper left corner.
https://www.serebii.net/events/dex/133.shtml Most recent Shiny eevee( if you want to argue it’s Shiny, I changed it to “shiny” on all my mentions of the LGPE event Shiny Pikachu/Shiny Eevee). It has a Cherish ball on its top left corner, showing it’s given in a cherish ball.
Check for the latest Shiny Eevee event in let’s go Pikachu. It’s obviously in a cherish ball. https://m.bulbapedia.bulbagarden.net/wiki/List_of_American_region_serial_code_event_Pokémon_distributions_in_Generation_VII#Eevee
I don’t care the bulbapedia link for Prhe Shiny Pikachu is Korean Region, the other links prove cherish balls exist in American versions too.
https://m.youtube.com/watch?v=aGnTXekscjI Minute 6 to 7 shows an eevee. Look closely at its discription it’s in a cherish ball it’s visible.