I am reading Bump’s book on Automorphic forms and Represenations and I am able to draw a lot of paralells between the theory of $GL(2, mathbb{R})$ which is the infinite place and the theory of $GL(2,F)$ where $F$ is a local field which corresponds to the finite places.

There are notions of smooth functions in both the setups. In the case of $GL(2,mathbb{R})$, the smoothness is the usual definition (considering the differential manifold structure on $GL(2,mathbb{R})$). But in the local case, smoothness is defined to be locally constant. Is there any differential structure on $GL(2,F)$ which justifies this definition? Or in other words, what is the motivation for defining smoothness in this fashion?