While I think that they definitely don't have a memory like ours, I've noticed that they do 'remember' when it comes to very basic, primal association.
The boys are fed Wardley betta pellets, it comes in a little blue food canister. For whatever reason we set ours next to Jeffery's cookie tub. A couple of days ago I was doing homework and started thinking, as well as watching Jeff. He's taken up 'begging', so he associates us with getting fed and has a way of showing us he wants to be fed. However I was curious how far all of this went.
So while he was swimming around, not paying attention to me all that much, I picked up the canister of his food. His fins lifted up and he immediately faced the food canister. I slowly dragged it around his tank, he darted towards, following it. I set it down. He started 'begging'. I did it for a few minutes, moving the canister around his tank, I even took our black dog pepper shaker that he runs away from usually and put the dog in front of the food canister. He flared! On whatever basic level I kinda think he gets canister=food, black dog=enemy, dog in front of food= >:(
Not a genius by any of our standards, but a step above goldfish lol.