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 proposed in the question, is everything pertaining to **nonlocal** phenomena.