Bipolar at that age is clinically rare although not unheard of. The best thing to do is to take her to a child psychologist. Talk therapy and cognitive behavioral therapy can be helpful. Family therapy as well. If they believe a further referral to a child psychiatrist is warranted they can then follow up.
Agreed, take your daughter to a psychologist. Our son was almost certainly among the rare kids who are ill with BPD in early childhood (age 3 1/2, to be precise, complete with what is now called 'pressured speech'), but back in those days, childhood bipolar disorder was considered impossible.
Having your child evaluated will almost certainly help all of you, and it may benefit her future in incalculable ways.
My daughter was very much like what you described. Even when she was a baby, she was either very irritable or very happy. She had a hard time snuggling, always pushing away when she was in my arms - she looked like she wanted to crawl out of her own skin at times. When she was a toddler, her tantrums were long lasting (several hours), with lots of kicking and anger, way past what a normal tantrum should be. As she got older, she would be out of control, running the house with her mood swings and demands (no one wanted to give her a hard time for fear of the consequences). And like your child, she would turn into a sweet kid after acting like a crazy child, wanting me to hold her and tell her that I still love her. It's like they know they were out of control and try to swing in the other direction for their own sake and our sake. She always had allergy shiners under her eyes (dark circles), with an on and off pale complexion. She had a major personality improvement when she went on a gluten free diet - the power of food! I'm not saying that your child has the same food problem, but the diet is worth a try. If it doesn't help, try eliminating other foods from her diet and see if it helps. Todays diets consist of sugar, white flours, soda, etc. All these things cause imbalances in the body (unless you have genetics from heaven - and most of us don't). Food allergies, food intolerances (lack enough enzymes to break down the sugar or protein in the food), crappy eating, etc. all effect our bodies and minds. All you have to do is clean up your daughter's diet and see what improvements follow. It's an amazing thing to watch. I wouldn't be in a rush to put a kid on a mood stabilizer without first trying basic diet changes. As kids grow, they need magnesium and calcium, etc. to feed their growing bones. If these minerals run low, irritability can result (just one example of many things that can go wrong). If you have any questions, just leave me a note. Hope this helps!
I haven't read all the answers thoroughly but we have an 8 yr old who is really similar in terms of horrible moodiness and swings of mood. Of course, me having bipolar I am always looking to see if my girls have any signs of any mental disorders in case it's genetic.
We have noticed over the past two years since she was 6 and became "the spawn of satan" that a lot of her moods seem to coincide with growth spurts. We notice she is "the devil incarnate", and then everything is ok again for a while and suddenly all her t-shirts and pants are too short. It's taken 2 years to see this pattern and of course it's not that simple but it helps us ride out the awful periods of about 2 weeks where I wonder where my little girl went, and I think she does too!
Lilbirds: I'm sorry that I don't really have any advice or input into your question besides agreeing with the other posters that the best way to tackle this issue right now is to get your daughter to a child psychologist who can refer you to a psychiatrist specializing in childhood disorders (if such a thing exists and you have access to it). I'm also so sorry that these very troubling things seem to be happening to your daughter at such a young age, and unfortunately pediatric bipolar disorder is a very controversial diagnosis right now, especially since very little is known about it. Seeing a mental health professional is an important first step in understanding what's going on in your child's mind.
Yarrow: I've recently studied a great deal about language development in children in a neuroscience/ psychology of language course, and I would be totally fascinated to hear about what pressured speech in a 3 1/2 year old entails. If you are willing to share some of the details about your child's language development (for example the rate at which your son acquired language, his mean length of sentences) and just how pressured speech presents in a person who is still in the early stages of forming semantic knowledge and a command of proper syntax... that would be so interesting. Because pediatric bipolar disorder is so rare and not well understood, it's very intriguing to hear that some children can present with pretty adult bipolar symptoms like pressured speech. I'm worried that asking these questions of you is going to come off as being cold and uncaring, but I really am so sorry that your child is already struggling with what could be a serious, lifelong disorder. I know this must be ridiculously painful and confusing time for the whole family. But hopefully working with a child psychologist will provide the tools and preparation to make it through. I'm just such a nerd and if you're willing to share, I would love to get a little insight into just what it's like to have such a relatively classic and adult presentation of manic symptoms in a very, very young child.