"As we approach the speed of light, everything around us slows down."
OK; yet it is not as simple as that, is it?
What do we mean when we say: "As we approach the speed of light..."
The speed of light relative to what?
For in our own individual frame of reference, the tool that we use to measure Spacetime from, we are at rest. We do not move relative to Spacetime, cannot know how or if we are moving as we have no way of determining or defining a state of rest for the Spacetime.
We can only measure our movement relative to other bodies - or rather, their movement relative to us - for we are at rest!
The only movement we know relative to Spacetime is that of light (electromagnetic radiation): because we know by experiment that light always travels at 'c'.
But how can that be? What does that mean? How can light move at the same speed relative to any observer?
Well the answer is very simple...
Light travels between Spacetime events. Locations in Spacetime at moments in time.
Each event is fixed and unique.
And, as the reference frame we are measuring from is at rest, as we measure it, in Spacetime, each event occurs at a fixed location, only the coordinates vary from frame to frame.
And as when we are measuring events there can be no movement (movement is change of location - and events are fixed - over time - and as events are fixed in time) and if what we are measuring is not moving there can be no time dilation or length contraction...
So light travels between events, whose only difference between frames of reference are their coordinates relative to the frame in question, Light always travels the same distance over the same time, regardless of the frame we use.
Therefore 'c' is the same whichever frame it is measured in.