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: $p$-adic numbers in physics
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.