Yes internet data travels at the same speed of light and sometimes even faster.
When we enter information on our computer and click send such as with this website where I'm answering this question.
The answer will be posted within milliseconds from my location to a data center that is likely thousands of miles away from me.
That's amazing that what I enter here on my computer with my keyboard will end up on a web server within milliseconds miles away from me once I click the add answer button.
The data I'm entering on here will travel really fast through many wires including copper wires, fiber optic cables etc and through many web servers and DNS servers until it reaches the proper server in the correct data center.
It's always amazed me that someone was able to invent the internet and have all this information travel through the air and through wires really fast.