reply to her, yes and mummy was wrong to do it, but you were wrong to bite my arm. lets try to be lovely to each other for the rest of the day. give her a huge cuddle, and then later on, at bedtime tonight, say to her how lovely she has been, but she must never bite, as it hurts the person who got bitten very much, just like her bottom hurt when mummy smacked it, and it is a a naughty silly girl who bites anyway,not lovely little princesses!!
not sure if this will help, but worth a try eh?
xx