I would like to know if they are not giving their lives after obtaining more than 6 trophies?

Hello @miimma and welcome to our wonderful King Community. I am not sure I understand your question. Lives are not dependent on how many trophies you may have earned. However when you rank up - which based upon your screenshot it looks like you are almost ready to go to rank 99 (as shown in the frame of your avatar) you will get 5 additional lives. The only thing you get with the trophies is to have them on your map and each time you earn more trophies your 'secret name' changes.