Tracking hits to my website with millisecond accuracy

I am conducting a user study in which users will be visiting a website that I manage and I need to record exactly when they visited the website. Tools like Google Analytics only give a general picture of traffic, but for this study I need to know exact times for hits.

Ideally the website would be hosted on GitHub pages, but if I need to do something like run a script to watch the site and keep a timestamp log then I could run it on a Linux server as well.