A lot of competition is dirty, and feminists have critiqued many aspects of male-dominated society, rightfully so. Feminists have been on the losing side of many male competitions. Also, women tend to be compassionate and opposed to a system in which someone loses. We know what it's like to lose . . . No, they inspired all the women they defeated to practice harder and achieve more themselves.
Many girls especially fear competing with friends, whether in sports or at work.
Girls do care about friendships. It's not right to tell her not to worry about her friends. She's going to worry about friends, and it's good that she cares. Tell your daughter that she can demonstrate excellence and competitiveness in a fair context that will help her friend succeed too. Encourage girls not to be afraid to win.