First thing your dd should have done is spoken to her form tutor: 'miss, you know when you kept me for 20 minutes into period 1 to talk about the tournament, well, Mr Bloggs gave me a detention for being late - could you email him & explain?' Sorted.
I would say it's vanishingly unlikely that that's how it went down, but if it did happen, that's that dealt with.
Assuming dd is in fact bang to rights for pratting about for 20 minutes & trying to pin it on the tutor...
Individual teachers don't usually just get to give out after school detentions on a whim. There's a system.
Where I teach, 4 lates or 4 behaviour points or a combo adding up to 4 gets you a detention.
Parents are notified, the child attends, 95% of the time that's the end of the matter.
Failure to attend = lengthier HT detention on a Friday.
Failure to attend that = not allowed back into class until a day in isolation has been served.
More than one isolation in a half term = not allowed back into class before a school/parent meeting has been held & a behaviour contract signed.
Breach of behaviour contract = invited to find a new school.
If a parent didn't want to support the school behaviour system, obviously we are not the right school for their child & we would advise them to consider other options.
So - once the poor form tutor has been eliminated from inquiries - I'd suggest a repeated request for detention on a different day, to allow for arrangements to be made.
If they won't budge or you won't, then the chips will just fall where they may. Internal isolation, probably. Most kids hate that & are careful not to land there again...