Let $X$ be a (non-empty) compact subset of $D(0,M):=\left\{x\in \mathbb{R}^n:\, \|x\|\leq M\right\}$, and let $f:X\rightarrow Y$ be uniformly continuous; for some metric space $Y$. Are there any known result guaranteeing that $f$ can be extended to a uniformly continuous function $F:D(0,M)\rightarrow Y$?

Let $X$ be a (non-empty) compact subset of $D(0,M):=\left\{x\in \mathbb{R}^n:\, \|x\|\leq M\right\}$, and let $f:X\rightarrow Y$ be uniformly continuous; for some metric space $Y$. Are there any known result guaranteeing that $f$ can be extended to a uniformly continuous function $F:D(0,M)\rightarrow Y$ such that the modulus of continuity of $f$ is (point-wise) greater-than or equal to the modulus of continuity of $F$?