It is your job, as a mother, to back your child up when you honestly believe she is telling the truth. Not to back her up just because she's your child (and believe me, many teachers have reason to believe parents simply side with their kids against teachers and staff on general principle, so I do understand where the teacher is coming from - especially if she still believes your daughter lied). The important thing is for adults to set a good example for kids.
I knew, growing up - and my kids have always known - that the only thing that will make me utterly furious is any form of fundamental dishonesty (lying, cheating, etc.) And if they do wrong, they come and tell me quickly - because they know that if they do wrong and lie to cover it up, or wait until I hear about it from others first, they're in ten times MORE trouble than they would be for doing something else wrong.
How unfair would the world seem, from a child's eyes, if all the adults backed each other up and sided against them - every time? It happens. I doubt they grow up to be happy adults, themselves. And on the flip side, what kind of message does it send a kid that mom or dad will angrily back them up, every time, even when they're actually lying through their teeth?
It's hard to be a parent. I won't even qualify that statement with "some days." Even on the easy days, you have to be vigilant - all the time. And children need adult advocacy, since they have little inherent power or authority.
It's hard to be a teacher, and they ALSO deserve to be backed up by parents whose kids do wrong - or might conceivably have done wrong. I just wish everyone could remember to follow the LAW - which, last time I checked, still said "presumed innocent until proven guilty."
I was accused of being a "pathological liar" by my kindergarten teacher. When asked what I was doing over the Christmas holidays, I truthfully answered that the family was going on a cruise to Africa. She scoffed and said "No one takes a five year old child on a trip to Africa!" and called my parents to discuss my lying. When they assured her I was telling the truth, she proceeded to exploit my terror of needles by telling me all about the vaccines I was going to have to get before the trip.
Back your child, but privately impress upon her how important it is for you to be able to trust her - how important it is that her behavior always be honest and respectful, because if you later learn she was lying, you'll be ashamed of having gone to the mat for her.
And do insist - and vigilantly watch for - fair and respectful treatment of her from this point forward, or switch schools. I ended up at a different school the next year, as my K teacher was also "promoted" to first grade.