I am assuming the binky is a pacifier.
We had the same issue. My daughter only got at naptimes and nighttime. She also did not sleep with it at school.
One weekend we went to my husband's uncle's house. We slept over. The next morning while we were downstairs the dog went upstairs and must have gotten my daughter's pacifier off of her toddler aero bed. He came downstairs with it in his mouth. It was very chewed up. My daughter (2.5 at the time) was laughing. We took it out of the dog's mouth. She gave it back to him. She thought it was a game. I made sure we took the remains home. When we got home and she got ready for her nap, she asked me for her pacifier. I had made sure we hid all of the extras. I told her that the dog had ate it. She looked at me like I was kidding. I went and got it. I showed it to her and repeated that the dog ate it. She got a very somber look. I asked her if she wanted me to hold her and she said no. She quietly laid down and was sad but never cried. Every day for the next week, we looked at the pacifier. Slowly she understood that the pacifier had been destroyed.
Every time we go over to the uncle's house she repeats how the dog ate the pacifier. She is now 3.5 years old.
FYI: We first tried cutting the ends off of the pacifiers but my daughter kept on sucking the pacifier. It did not affect her at all.