Sums of squares in function fields over Henselian local fields

We give upper bounds for the level and the Pythagoras number of function fields over fraction fields of integral Henselian excellent local rings. In particular, we show that the Pythagoras number of $\mathbb{R}((x_1,\dots ,x_n))$ is $\le 2^{n-1}$, which answers positively a question of Choi, Dai, Lam and Reznick.