I will start with an example and then discuss a potential example that turns out to be (most likely) a non-example.
An example is the fractional Gaussian field with sufficiently large Hurst parameter. Informally, the fractional Gaussian field $h$ on $\mathbb{R}^d$ with index $s \in \mathbb{R}$ is given by $h = (-\Delta)^{-s/2} W$, where $(-\Delta)^{s/2}$ is the fractional Laplacian on $\mathbb{R}^d$ and $W$ is a white noise on $\mathbb{R}^d$ (the usual Gaussian free field is the case $s = 1$). A rigorous definition, as well as a comprehensive survey of this class of processes is given in this paper by Lodhia, Sheffield, Sun, and Watson.
The Hurst parameter is defined by $H = s - d/2$. If $H > 0$, then $h$ lives in a quotient of the space $C^{\lceil H \rceil - 1}(\mathbb{R}^d)$ of $(\lceil H \rceil - 1)$-times continuously differentiable functions. In particular, an example of a model with $C^1$ level sets is the fractional Gaussian field with $H > 1$. Regarding the correlation functions, the regularity of a Gaussian field is closely related to the regularity of its covariance; in the case of the fractional Gaussian field, see Theorem 3.3 of the aforementioned paper.
Going a bit further, we might ask whether the fractional Gaussian field arises as the scaling limit of models that are not a priori Gaussian. A natural example to consider is the Ising model with interaction $J_{x,y} = (-\Delta_{\mathbb{Z}^d})^s_{x,y}$ for $x, y \in \mathbb{Z}^d$ with $\Delta_{\mathbb{Z}^d}$ the graph Laplacian of $\mathbb{Z}^d$ (when $s = 1$, $J$ is a nearest-neighbour interaction).
Whether or not the scaling limit of this model is (expected to be) Gaussian depends on the dimension $d$. Precisely, it is believed that there exist a number $d_c$ (not necessarily an integer) such that the scaling limit is Gaussian if and only if $d > d_c$. In this case, I would expect the scaling limit to be a fractional Gaussian field with index $s$.
When the interaction is given as above, all evidence points to the fact that $d_c = 4 s$. See, for instance, this paper of Heydenreich, van der Hofstad, and Sakai (in this paper, $\alpha = 2 s$ in the notation I am using). In particular, if $d > d_c$ then $H < 0$ and the predicted Gaussian scaling limit cannot be defined as a continuous function, so this turns out be a non-example.
