I can tell this is really hard on you. You were trying to do the right thing and it seems to have back-fired. Instead of feeling better, your little girl seems to feel worse. I think she is a very bright 3-year-old. She may be young to be dealing with this but she’s letting you know you can’t avoid it. She isn’t willing to be put off with promises of heaven or to put the question aside. She is grappling with the big questions around love and loss and meaning.
I have an idea that maybe she doesn’t think you understand her distress. Instead of arguing with her, however gently, why not just hold her and agree that it is indeed very, very sad that people and animals and every living thing come to an end. Let her cry it out. Soothe her but don’t contradict her or try to make it all right. Tell her that you get sad about it too.
When she settles down, tell her she can always come to you when she is sad. Then see what happens. At that point, she may be more ready to hear that we hold people we love in our hearts and they never really go away; that our job is to make lots of good memories so we can carry on.
There are lots of good children’s books about death that might be helpful. Talk to your librarian for suggestions or do a search online. Hearing some stories about how other children managed these big ideas may give your daughter new ways to handle it.
I wish you well.