I think (I have never tried so don't really know it) it's possible. However there are some important things to consider - most calculations are floating point operations, so you either need FP code, or find a way to use integer operations with enough precision.
Next, depending on what you need to do the PIC could be pretty slow (to train the net). If you just want it to use already trained net it could work.
Regarding the PIC type, probably the best bet would be the 18F series, but maybe you can do something smaller in 16F as well (with the above things in mind)
BR