  In the present paper, we generalize the well-known Hensel's lifting lemma to
any continuous function $f : \mathbb{Z}_p\rightarrow \mathbb{Z}_p$. This
answers a question posed by Axelsson and Khrennikov (2016) who showed the
validity of Hensel's lemma for $1$- and for $p^\alpha$-Lipschitz functions. For
the statement and the proof, we introduce a suitable generalization of the
original van der Put series. We use the concept of approximability of
continuous functions to give numerical examples.
