Given a field, $K$, are there two DVRs $A,B$ for the same field $K$, such that $A\cap B$ is also a field?

Question

Clearly, we cannot construct any such DVR's when $K=\mathbb{Q}$, or when $K=\mathbb{F}_q$, so we must look to more intricate examples. Is it possible to construct such DVR's when $\mathrm{char\,} K = 0$? My intuition tells me that we should consider positive characteristic, but I can't figure out how to construct anything like this.