Hang in there, it will get better and eventually she will stop asking for it. We had to have the "Binky Fairy" come visit our daughter just a few months before her 3rd birthday, told her it was going to be given to a baby that really needed it, and she kept asking for it to come back for about a week - "But it is MY binky! I want it back!" But as the week went on, she got less tearful and dramatic about it, and finally stopped altogether. My DH was tempted to give it back but I told him no way - we've told her it's gone, and we have to have her believing us that what we tell her is true, and doesn't change.
No, you are not causing psychological damage. They all have to get over it at some point. Better this than having her still needing it at 5 or 6.
It has been almost 2 years now. When I tell DD all about how she used to have a binky, she laughs and tells me, "But Mom! Binkies are just for babies!"