This is food aggression and resource guarding.
How long have you had Lilly?
What rules and boundaries does she have to abide by?
Dogs who do this normally see themselves as the pack leader,therefore taking control of what they deem necessary.
I suggest you put her on a strict Nothing in Life is Free program. (Email me at
training@libertydogtraining.com and I'll send it to you! Just make sure you put YT forum in the subject line so I'll know you are from here!) This will safely establish yourself as the pack leader, and help diffuse this situation safely.
If you see the puppy go after her food (even if she dropped is away from the bowl) correct the puppy by saying "no" and bringing the pup back to her own dish. This will show Lilly that you are in control, not the puppy, and not her, so she will not have to "take control" because she will start to feel more comfortable knowing you will.
If you can't supervise them two together around toys or food, then put the puppy in a crate safely away from Lilly. There is ALWAYS a pack leader. If you leave the room, Lilly will step up and take your place while you are gone. This might mean bad news for the puppy if left alone.