TITLE: 'Important' applications of p-adic numbers outside of algebra (and number theory).
QUESTION [54 upvotes]: Surely, $\mathbb{Z}_p$ and $\mathbb{Q}_p$ (and their extensions) are very important for algebra and number theory. Do they have any important applications outside of algebra (that I could easily explain to a student)? Here I do not demand the applications to be (purely) 'mathematical'; for example, I wonder whether p-adic numbers have applications to physics (outside of string theory?). Moreover, I am also interested in those applications that are partially 'algebraic', and yet important for some other parts of mathematics.

REPLY [4 votes]: In my answer to this related MO question, I tried to explain the utility of using $\mathbb{Q}_p$ based toy models in order to better understand questions in mathematical quantum field theory in physics.