  First-principles calculations of the essential spin-orbit and spin relaxation
properties of phosphorene are performed. Intrinsic spin-orbit coupling induces
spin mixing with the probability of $b^2 \approx 10^{-4}$, exhibiting a large
anisotropy, following the anisotropic crystalline structure of phosphorene. For
realistic values of the momentum relaxation times, the intrinsic
(Elliott--Yafet) spin relaxation times are hundreds of picoseconds to
nanoseconds. Applying a transverse electric field (simulating gating and
substrates) generates extrinsic $C_{2v}$ symmetric spin-orbit fields in
phosphorene, which activate the D'yakonov--Perel' mechanism for spin
relaxation. It is shown that this extrinsic spin relaxation also has a strong
anisotropy, and can dominate over the Elliott-Yafet one for strong enough
electric fields. Phosphorene on substrates can thus exhibit an interesting
interplay of both spin relaxation mechanisms, whose individual roles could be
deciphered using our results.
