A.M.
why stop playing to eat? seriously its hard to sit and eat sometimes haha. the first thing i do is cut out juice and sweets when my kiddos don't eat because the sugar in those items make the stomach feel full and decreases appetite. at three they like to show their control make up their minds and such. get her to participate in preparing the food like melon/grape/apple scewers, banana smily face pancakes/waffles, english muffin pizzas. small amounts of food help my kids to snack more often which is good snacking is what kids do their portion sizes are small because they are small. i worry more about WHAT they are eating not how much. you could make a list of the foods she tells you what foods she likes when you designate snack time she can pick what she wants when it is meal time have her serve herself or help pick out the side dishes ie:"kid" do you want peas or carrots with dinner? or grapes or oranges? if she only eats a bite or two well so be it.
try to space out your snacks/meals a couple hours apart. as long as she's getting some good food she'll be fine. if you make it a big deal she'll know she's frustrating you :P good luck