Your daughter sounds a LOT like my son. He is now nearly 4, and will balk me at every turn when I try to introduce academics to him. In comparison, my daughter, who is now nearly eight, could read by the time she was 3 1/2 and most of her ability was absorbed from reading together and occasionally asking her to "help" me sound out words. A lot of her ability to read at that age was simply that it interested her to do so.
I found it comforting to read that in (somewhere in Scandinavia... am blanking now...) they don't start teaching kids to read until they are 6 or 7 and they have on of the highest literacy rates in the world.
I know from experience that trying to coax or pressure or force my son into playing "school" with me will result in resistance. We expose him to phonics concepts throughout the day and don't ask for him to do anything with it. He collects the information in his head and occasionally "spits" it back out in ways that surprise us. When the mood strikes him, he will sound out words in the environment (a colorful billboard, a book cover with a picture that intereste him, words on a t-shirt), and I have been pleased to find that if there are short vowels and no digraphs that he actually CAN read a few words. Not that you would know it if you try to work with him.
I honestly think that there is too much pressure for kids to read early, although, like much of education theory there is NO evidence to support early reading with improved learning or ability to read better down the road. I DO believe that our schools have ridiculously low standards and that we are churning kids out of our public school system who can barely read and who can't reason at all. But I DON'T think the answer is to encourage earlier and earlier reading. In fact, it may be counter productive, if they are burned out on school before they even start.
At age three, kids naturally enjoy exploring ALL kinds of things around them, and all the things they learn at this age will stimulate their brain and their curiosity. If they love learning new things, eventually that love and curiosity will fuel their interest in reading (or de-coding the secret language of books, signs, and secret notes!). Please don't rush your little girl... if she is showing that she does know the answers but is refusing to answer, I think it's a clear signal that she's feeling very stressed by situation and wants to stop.
Anyhoo - best of luck and I hope I didn't sound too preachy!