  In this paper, using the tools from the lineability theory, we distinguish
certain subsets of $p$-adic differentiable functions. Specifically, we show
that the following sets of functions are large enough to contain an infinite
dimensional algebraic structure: (i) continuously differentiable but not
strictly differentiable functions, (ii) strictly differentiable functions of
order $r$ but not strictly differentiable of order $r+1$, (iii) strictly
differentiable functions with zero derivative that are not Lipschitzian of any
order $\alpha >1$, (iv) differentiable functions with unbounded derivative, and
(v) continuous functions that are differentiable on a full set with respect to
the Haar measure but not differentiable on its complement having cardinality
the continuum.
