In this paper, we introduce curve-lifted codes over fields of arbitrary characteristic, inspired by Hermitian-lifted codes over $\mathbb{F}_{2^r}$. These codes are designed for locality and availability, and their particular parameters depend on the choice of curve and its properties. Due to the construction, the numbers of rational points of intersection between curves and lines play a key role. To demonstrate that and generate new families of locally recoverable codes (LRCs) with high availabilty, we focus on norm-trace-lifted codes. In some cases, they are easier to define than their Hermitian counterparts and consequently have a better asymptotic bound on the code rate.