Modular periodicity of binomial coefficients

We prove that if the signed binomial coefficient $(-1{)}^i\left(\genfrac{}{}{0px}{}{k}{i}\right)$ viewed modulo p is a periodic function of i with period h prime to p in the range $0\le i\le k$, then k+1 is a power of p, provided h is not too large compared to k. (In particular, $2h\le k$ suffices.) As an application, we prove that if G and H are multiplicative subgroups of a finite field, with H<G, and such that $1-\alpha \in G$ for all $\alpha \in G\setminus H$, then $G\cup \{0\}$ is a subfield.