For a real number r≥0, we define r-fold differentiability of a function on a p-adic vector space by the convergence of its Taylor polynomial expansion, and compare this differentiability definition with that by iterated divided differences, the textbook approach (from the 80’s) to define p-adic differentiability.This comparison applies to a recent definition of r-fold differentiability over a p-adic number field K that arises from the p-adic Langlands program over GL2(K); yielding that this differentiability condition is equivalent to that via divided differences on K as vector space over Qp.