Let $\mathbb{S}^n$ be the $n$-sphere. I would like to know if anyone knows of the following result in the literature (or whether anyone knows a proof/counterexample).

Let $f\colon\mathbb{S}^1\times[0,1]\to\mathbb{S}^2$ be a $C^1$-function such that $f(x,0)\ne p$ for all $x\in\mathbb{S}^1$. Then there exists a continuous function $g\colon[0,1]\to\mathbb{S}^2$ such that $g(0)=p$ and $f(x,t)\ne g(t)$ for all $(x,t)$.

The existence of space filling curves necessitates $f$ being a $C^1$-function, but other than that, I have not succeeded to construct a counterexample or a proof strategy/idea.