Although the use of elephant ivory has been banned in most western countries, there are still countries where it is being used, fueling the poaching for ivory. If the instrument is made in USA or Canada, or the pins are made and packaged in those countries, you may be certain that the ivory is "fossil" and came from skeletal remains of animals that died natural deaths a long time ago.
Permafrost conditions are such that some mammoth tissues and hair have been preserved, along with the bones and tusks. As John in Kansas has pointed out, the mastodons lived until fairly recently, when they were wiped out by hunters about 10000 years ago.
Another type of material with a much longer preservational life is the exine, or covering of plant pollen and spores, and some unicelled animals. These materials are used by palynologists to date sedimentary rocks as old as 500 million years. The rock itself is dissolved in hydrofluoric and other acids to free these remains which are concentrated and studied under the microscope. Palynologists started working on these remains in the labs of the major oil companied some 60 years ago, building on scattered academic studies going back about 130 years. Although chemically changed, these materials have retained sufficient organic characteristics that they are not mineralized or "fossilized" in the usual sense.