Fiberous Material White Material below Plaster - Asbestos?
I am almost certain that it is  horse hair.  It was added to plaster as a binder in the old days 
This site gives a history of its use and answers a ton of questions you might have about it.   

For starters :


  Horsehair plaster, also known as lath and plaster, is typically found in older homes and homes of significant historical value. This practice is not commonly used today. In fact, drywall began replacing the process in the 1950s. However, you can still find a lot of homes with horsehair plaster.
  If you’ve never heard of horsehair plaster before, then you’re probably pretty confused. So, what is horsehair plaster? Why is it called horsehair plaster?
  Although there are different variations, the most common type of horsehair plaster is a mixture of lime, sand, plaster, and horsehair. Yeah, that’s right, horsehair. The horsehair that was used was from the mane and the tail of the horse. Horse hairs are long, strong, and when they’re mixed with plaster, they offer a lot of structural strength.


Still if you want to be positive it is not asbestos have it tested.  

And if you need something else to worry about there is a possibility of  anthrax contamination in old horse hair plaster!  

Yikes!