Since the product of the elements of $K^*$ is $-1$, if $A$ and $B$ form a partition of $K^*$, then $(\prod_{a \in A} a) (\prod_{b \in B} b) = -1$, so $\sum_{a \in A} a = \prod_{b \in B} b$ if and only if
$$\left(\sum_{a \in A} a\right) \left(\prod_{a \in A} a\right) = -1. \quad (*)$$
Let $|K| \ge 4$. If the characteristic of $K$ is $2$, then the singleton set $A = \{1\}$ clearly satisfies $(*)$. If the characteristic of $K$ is odd, choose an element $a$ in $K^* \setminus \{\pm 1\}$, and notice that the 3-element set $A = \{-1, 1, a\}$ satisfies $(*)$.