Upvote:5
Many of the bigger tourist beaches have public shower heads (in the open air) where you can rinse of sand and suntan lotions, but in addition to that some, including the beach at Playa de Palma
https://www.seemallorca.com/beaches/playa-de-palma-beach-majorca-palma-de-mallorca-661001
Have small toilet and shower blocks located on the promenade.
There is also a shower room at Palma airport.