Does a shiny pokemon stay shiny after evolution?

Yes, a shiny pokemon will remain shiny after evolution.

Upon evolution, a Pokémon will retain its Shiny status. A Shiny Charmander, if leveled up, will eventually become a Shiny Charmeleon and then a Shiny Charizard, just as a regular Charmander will become a regular Charmeleon and then a regular Charizard. This is due to the status being determined by values inherent to the Pokémon.

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If a shiny pokemon evolves, it will stay shiny. For example, if you have a shiny sentret and evolve it into furret, it will stay shiny. I am certain because my shiny azurill evolved into marill and azumarril, still being shiny.