Well if you take out partial derivatives, at least quantum field theory and in particular conformal field theory will survive the massacre. The reason is explained in my MO answer:
https://mathoverflow.net/questions/259155/p-adic-numbers-in-physics/259160#259160

One can use random/quantum fields $\phi:\mathbb{Q}_{p}^{d}\rightarrow \mathbb{R}$ as toy models of fields $\phi:\mathbb{R}^d\rightarrow\mathbb{R}$. In this $p$-adic or hierarchical setting, Laplacians and all that are nonlocal and not given by partial derivatives.

Most equations in physics are local and therefore need partial derivatives in order to be formulated. What should remain, in the very hypothetical scenario in the question, is everything pertaining to nonlocal phenomena.