Many Christians interpret human creation as merely spiritual. Most believe God gave us the gift of freewill and a soul. It is possible to take it a step further and believe that God literally made us just like him with all his knowledge and power. Many religons speak of reaching elightenment and science is always trying to expand our knowledge of the human potential. Some rare people may have tapped into these abilities and have done miraculous things that can not be explained.
If you understand what Jesus was saying in the bible you have to come to that conclusion. With enough knowledge you can do incredibly "God-like" things. I'm not saying people can do biblical sized miracles but maybe one day. We only see a fraction of what we are really capable of.
We don't have godlike power. I'm sure any sane person wouldn't give that amount of power to an entire race of creatures. Can you imagine the amount of destruction that would cause? This is why when we are creating artificial life we have to make sure that we put some precautions in place so that they don't surpass us in power.
I guess you could see our intelligence as a sort of power compared to other creatures in this world. Who would have thought that puny, weak humanity would become the most powerful creature on the planet? We've used our technology to elevate ourselves to the point where we basically rule most of the world. So I'd say our intellect is our power.
We don't have the powers of God because God created us, he could have created us with powers, but he created us without powers.
The people that the news sometimes talks about having done supernatural things are not powers from God, but powers from Satan if what the Bible says is correct, which it probably is.