No. The speed of light is a constant. This means distance and time have to be variable.
Let's take your experiment to extreme. A man riding a spacecraft at 5km/hr slower than the speed of light. He passes a man that flashes a light in the direction he is traveling. With your point of view he should be able to reach out and grab one of these photons traveling past him at 5km/hr. Relativity says the photons will actually pass him at the speed of light.
So both the man in the craft and the man with the flashlight will measure the photons traveling at the speed of light. (note that due to the Doppler shift, the frequency of light for the man in the craft will be near zero Hz) For this to happen, both men measure the time it takes for a photon to travel a certain distance. They both come up with the same time and distance. This means their measure of time and distance is not the same.
Look up Lorentz transformation. The equations may look tough at first glance, but the denominator (the square root of 1-v^2/c^2) is simply saying as velocity v approaches speed of light c, the fraction v^2/c^2 approaches 1. So as you approach the speed of light. the denominator gets smaller. think it out. it will make sense.