A quick Google turned up
THIS place.
FWIW I stayed near Puerto del Carmen a couple of years ago, but I took my own bike. Never the less, I can recommend
PRO BIKE in La Santa as an LBS. Granted it's not exactly convenient from Playa Blanca, but the island is only small so nothing is really that far away.
Warning, riding out of La Santa back to Play Blanca via Mancha Blanca and Yaiza will involve an 8km climb
The riding is fantastic though, the roads are far better than ours, and the drivers give you loads of room. They're used to cyclists from the Ironman and also a fair few pro teams use the island for winter training
The only thing to watch for is the volcanic aggregate they use, it's sharp as hell and eats tyres for breakfast
