My fluid mechanics is not strong, but a steep increase in the pressure from a beam that you are heading towards as you approach the speed of light does indeed happen.

You can reason this from several perspectives.

Suppose you measure a plane wave in one frame and find out that the electric and magnetic field are $E_0\,\cos(\omega_0\,t)\,\hat{X} $ and $H_0\,\cos(\omega_0\,t)\,\hat{Y}$ where you are facing the $-\hat{Z}$ direction. Now you begin to move in the $-\hat{Z}$ direction at speed $v$. If you look up the Lorentz transformation for the Faraday tensor, you find that you now measure the following electric and magnetic fields:

$$E_0\,\sqrt{\frac{c+v}{c-v}}\,\cos\left(\sqrt{\frac{c+v}{c-v}}\,\omega_0\,t\right)\,\hat{X} $$ $$H_0\,\sqrt{\frac{c+v}{c-v}}\,\cos(\sqrt{\frac{c+v}{c-v}}\,\omega_0\,t)\,\hat{Y}$$

Notice that the relativistic Doppler effect is included here, and amounts to a blue shifting of the light whereby the frequency is scaled by $\sqrt{\frac{c+v}{c-v}}>1$.

Therefore, if you're holding onto a mirror, the initial pressure will be $P_0=2\,S_0/c = 2\,E_0\,H_0/c$, and when you begin pushing it towards the source the pressure will rise to $P_0\,\frac{c+v}{c-v}$. As you can see, this diverges as you approach the speed of light relative to your initial frame. So this does bear a vague likeness to the character of a greatly increased force as one approaches the speed of sound.

You can also think in terms of photons. The photons are blue shifted, so that each one's energy is raised by a factor of $\sqrt{\frac{c+v}{c-v}}$ since their energies are proportional to their frequencies. At the same time, the frequency of collision with them also undergoes a Doppler blue shifting (you collide with them oftener), so the rate of impulse transfer to you also scales by the square Doppler factor and again we find $P_0\,\frac{c+v}{c-v}$ as the light pressure on you from the beam.

What's interesting about this analysis is that the two answers only come out to be the same if and only if the postulated photon's frequency is proportional to its energy. No other dependence will work - so, as I argue in my answer here, there are already some hints in the classical Maxwell's equations and classical special relativity about what the energy relationship for the quantum photon ought to be: they foretell a Plank's constant without fixing its value.

A full answer to your question would also deal with the analysis of a medium: evidently there is no infinite divergence as a body passed through the medium at the speed of light in the medium, and Cherenkov radiation is a real, verified effect. However, right now I can't complete the answer because I have not thought about this before.

In closing, one should also note the following interesting facts about this kind of thought experiment:

1. The Celestial Sphere is mapped by a boost in the observer's frame by rapidity $\eta$ in the same way that the Riemann sphere is mapped under the Möbius transformation $\zeta\mapsto e^{2\,\eta}\,\zeta$. Look up the "Lorentz Transformations section on the Wikipedia Möbius transformation page. This effect is known as the relativistic headlight: the whole night sky becomes squashed into a deeply blue-shifted region ahead of you, whilst the rest of the sky becomes a small region around what's behind you greatly "zoomed up" (stretched over the sphere) and red-shifted;

2. The xkcd "Relativistic Baseball" whatif article and "Diamond", although about different effects, are obligatory reading here as they highlight some of the issues arising for objects travelling at very high speed relative to the matter around them;

3. In practice the divergence in pressure from the light from any particular source won't be quite as bad as that foretold by the $\frac{c+v}{c-v}$ factor: the above assumes a plane wave which is infinite in extent. In particular, as you move very swiftly towards a source, its distance from you becomes deeply Lorentz contracted, so you "scoop up all its energy" very swiftly, pass the source and then (as long as you avoid the collision) it's light is either redshifted (if it's radiating in both directions) or extinguished. A full calculation therefore needs to account for the full Fourier transform of the transient light signal.