This perhaps elaborates on Greg's answer regarding quantum field theory on the p-adics. One approach to studying "regular" quantum field theory over $\mathbb{R}^d$ is to consider probabilty measures on the space of fields $\phi:\mathbb{R}^d\rightarrow\mathbb{R}$. These are obtained as limits of Radon-Nikodym pertubations of better understood Gaussian measures. The mathematical analysis involved, typically based on rigorous renormalization group techniques, is very difficult. One can define analogous models over the p-adics where the fields $\phi$ become random generalized functions $\mathbb{Q}_p^d\rightarrow\mathbb{R}$. There the analysis is much easier, and therefore the p-adic case is a good "toy model" for the real case. The idea is to develop one's tools and methods on the p-adics first, and then add the necessary refinements to handle the real case as a second step. Most of the core difficulties of the renormalization group analysis are already present in the p-adic toy model. However some features of the "regular" real case such as the flow of the wave function renormalization coupling are specific to the real case and therefore cannot be studied with this toy model. This being said I also believe such models over the p-adics are worthy of study per se and not just as toy models for "regular" QFT over the reals, in which I agree with Theo's comment above.

This perhaps elaborates on Greg's answer regarding quantum field theory on the p-adics. One approach to studying "regular" quantum field theory over $\mathbb{R}^d$ is to consider probabilty measures on the space of fields $\phi:\mathbb{R}^d\rightarrow\mathbb{R}$. These are obtained as limits of Radon-Nikodym pertubations of better understood Gaussian measures. The mathematical analysis involved, typically based on rigorous renormalization group techniques, is very difficult. One can define analogous models over the p-adics where the fields $\phi$ become random generalized functions $\mathbb{Q}_p^d\rightarrow\mathbb{R}$. There the analysis is much easier, and therefore the p-adic case is a good "toy model" for the real case. The idea is to develop one's tools and methods on the p-adics first, and then add the necessary refinements to handle the real case as a second step. Most of the core difficulties of the renormalization group analysis are already present in the p-adic toy model. However some features of the "regular" real case such as the flow of the wave function renormalization coupling are specific to the real case and therefore cannot be studied with this toy model. This being said I also believe such models over the p-adics are worthy of study per se and not just as toy models for "regular" QFT over the reals, in which I agree with Theo's comment above.

Edit: I gave a few more details about this particular toy model in this MO answer. Also note that there has been quite a bit of action going on recently in this area since $p$-adics provide a simplified setting in which one can study the so-called AdS/CFT correspondence. See, e.g., Gubser's talk at the Strings 2016 conference.