Why are p-type solar cells more prone to degradation in space than the n-type ones?
The space radiation environment is a quite complex subject. Generally speaking one needs to be careful about just where in space you are concerned about - near earth orbit (within the Van Allen Belts), outside the belts, regularly passing through the belts, interplanetary, etc. 

That being said, the three main areas of concern are protons from the solar wind, electrons from the solar wind, and cosmic rays from the rest of the universe. For testing purposes the standards for the solar wind are 10MeV protons and 1MeV electrons. I will focus on these for the purposes of this answer. 

As the protons or electrons hit a solar cell, the will go in some ways and lose energy through interactions with electrons and the nuclei (electronic and nuclear stopping, respectively). Electronic excitations at best make a little more current in the depletion region. The interactions with nuclei are more problematic - either particle can transfer enough energy to a nucleus to displace it off a crystalline lattice site (protons are about 1000 times more effective at it then electrons because their mass is much higher). So, as the cell accumulates dose, more and more atoms are off their sites. 

OK, what does that do? Well, point defects in semiconductors often create mid-gap electronic states that allow carriers to recombine - your cell is now less efficient at making current since some gets lost. Further, point defects (vacancies and interstitials) can combine with either other point defects or impurities to form more complex defects, and these may be even better recombination centers. 

Of particular relevance for your question is the impact of boron. To quote from 
T. Hisamatsu et al. Radiation degradation of large fluence irradiated space silicon solar cells (1998):


  Radiation-induced defects associated with boron atoms are said to form
  recombination centers for minority carrier electron and this phenomenon is said to
  cause the degradation in low-fluence regions. Therefore, using highly boron-doped
  substrates in order to avoid the sudden drop of cell performance should not be
  a proper means to provide cells with higher radiation tolerance.