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there are a number of different places in that area, I am sure one of them would do a days hire but perhaps need some notice.
Other wise you could try a cycle store in Santa Catalina (about 15mins on the train from Palma) but i cannot remember the name i am afraid.
its actually not a bad place to start and finish from so you could catch the train out in the morning, pick up the bike and head out. it would save you navigating your way into the city.
thats all i got i'm afraid!
Yeah the place I used last year was in Can Pastilla but it seems they don't do 1 day hire. Ha, yep I've actually had a night out in Arenal, it was funny. Will keep looking....