That depends on how you define memory. If for water it is defined as the property to have a stable state for sometime, then it has memory, just not a very good one – 50 femtoseconds is its retention time. That’s about 60 million million times shorter than the mythical goldfish’s three-second memory.
But with that “memory”, water could not retain any useful information. The memory is just its ability to form an ordered group of water molecules that can last for 50 femtoseconds. It is a bit like a crowd of people all milling around in train station, there are pockets of order where people are standing around looking at departure boards or getting a coffee. But these groups will disperse after a while. And so it is with water, there are pockets of order where the water molecules are interacting with each other and with things that are dissolved in it, but these are lost pretty quickly.
Let’s try another question. What if water had an elephant’s memory and never forgot?
In that case all the ordered pockets would hang around forever. But it wouldn’t look much like liquid water anymore. Instead it would be quite different, in fact you would probably call it ice.
How about we try something a bit more bizarre?