Our 9 year old daughter is a perfectionist. She will burst into tears and say she is a "loser" for getting 2nd or 3rd place, even if she's beat many other people or teams. She is also like this if she gets anything less than an A. She is intellectually gifted and an amazing athlete, so she hasn't experienced failure very often. We validate that she is disappointed, but her response to getting anything but 1st place is bad sportsmanship, though she has fairly good sportsmanship the rest of the time.
Unless it's excessive, being competitive is not a bad thing, but being a bad sport is. Your daughter should want to succeed, but not at the expense of becoming unhappy with herself or being unpleasant around others. Talented kids often set high expectations for themselves, but this often reflects parents' reactions to the kids' accomplishments. This situation deserves an exploration of the root causes.
Reevaluate your own responses to her school and sports achievements. Ask yourselves whether your positive reinforcement is excessive and whether you are both being sincere in your discussions with her about being satisfied with finishing in 2nd or 3rd place. As parents, do you handle your own "second place finishes" or "B" grades well? Are you driven to personally be at the top of everything you do? If you are not the perfectionist influences in her life, explore who else might be. Are there other kids who are pushing her by being uber-competitive? Is she in accelerated classes where the teachers are driving the kids?
In the meantime, encourage your daughter to participate in non-competitive activities that you enjoy as a family or that she can enjoy with friends. Look into after-school clubs, charity and volunteer work, hiking and camping. The more fulfilling she finds non-competitive endeavors, the more she'll appreciate fun over winning.