No. That is your normal state of affairs: when you're "sitting still" in space, you're "moving through time" at the speed of light. I'm using scare quotes here because, of course, there is no such thing as "sitting still in space" because all motion is relative. The point here is that "light is always moving at the speed of light relative to you (no matter how you're moving)" is the same as saying "you are always moving at the speed of light relative to light (no matter how you're moving)". The way that works is, as I said, that this constant speed is a measure of your motion through spacetime, not through space alone or time alone.