ANSWER: Yes, however only for Enterprise level clients. Git is an awesome technology that is by far the most popular way to track and manage versions and updates amongst highly skilled web developers. However, because of its complexity and high learning curve, it typically requires some sort of serious team or in-house group of developers to make it worth the while (and to avoid totally screwing things up).
For thing reason, only companies with professional developers experienced in Git should be using it. And because of the extra support etc required, we only offer it currently to our Enterprise level clients.
We install Git on your server, and encourage you to make use of it via VersionPress