There are currently 3 reasons why you are not able to earn SLP in the game.
Using a recently gifted Axie
An Axie that was gifted from another account will not be eligible to earn SLP from arena for 24 hours after gifting. Just having one recently gifted Axie on the team will make the whole team ineligible.
Using a recently bought Axie
An Axie that was bought from the Marketplace will not be eligible to earn SLP from arena for 24 hours after they were bought. Just having one recently bought Axie on the team will make the whole team ineligible.
No Energy
You will not be able to earn any SLP from winning an arena match if you do not have any more energy left on your account.