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For the first time, Pokémon Mystery Dungeon Rescue Team DXincludes Shiny Pokémon. These Pokémon will rarely appear in various dungeons as powerful bosses or in general and can require a lot of damage to be taken down. Normally they are just standard colours but you have a small chance of encountering them shiny. On the map, these Pokémon are marked with a yellow “Strong Foe” symbol
When taken down by your leader, there is a slim chance they will join you. However, if your leader has a Friend Bow equipped, there’s a much greater chance they will join your team.
Only 27 Pokémon have the ability of being Shiny.
Source
Tl;Dr: Yes.
The list of Pokemon that can be shiny are
Hope this helps!