When I was younger (I am 62) children were smacked for crossing red lines. Some parents overused physical punishment but it was generally accepted that a small smack for a bad misdemeanour - or endangering behaviour like running into a road or putting fingers into sockets or on cookers - provided it was for the child's benefit and not to indulge the parents' sadistic or vengeful impulses, was a necessary evil.
Everyone accepted this and I know of no single rational adult who ever grew up and resented their parents. In fact, I can recall many who have acknowledged they were very naughty and would not have behaved had not their parents had an ultimate sanction. It was always thought that a short, light smack on the back of the legs was better than a child dying in the road. The fact that a sanction was there meant it was rarely resorted to.
However, over time all forms of physical punishment have become anathema and unacceptable. I can see that there are arguments for this as some parents would have taken it too far and there might be some truth in the 'you shouldn't meet violence with violence' argument - though you could also argue that a small warning tap does not constitute 'violence' if well meant.
Personally, while I would never go against the law in such matters, I think that alternative punishments such as emotional coercion could be as bad or even worse and punishments which go on and on rather than immediate and tied to the misdemeanour in the way a small tap is, are soul destroying. They are also pointless when children are very small and cannot conceptualise an abstract continuing punishment being consistent with a sudden burst of naughtiness.
I think we have all got our knickers in a twist about this issue. I find it very sad that a loving, caring mother whose ten year old daughter has kicked her (which is off the Richter scale in terms of poor behaviour) should be so wracked with guilt that she is considering reporting herself to the SS for reacting in a totally understandable way. The reaction was immediate and without premeditation and will certainly make the child think twice about her behaviour. In the grand scheme of things it will have had no adverse effect on her daughter unless the child is so badly behaved she uses the mother's guilt against her.
This last contingency is quite likely in the climate where children are indulged and parents do not support each other in matters of discipline for the children's own good. After all, no spoilt, selfish, mannerless child who takes these qualities unchallenged into adulthood will be employable or make and keep friends.
OP, there will be no lasting ill effects from this incident. It's a one off. Do not let it wrack you with guilt.