Unique fixed point of contraction defined on a ball

Question

In the case where $f : X \rightarrow X$ is not a contraction on the whole space $X$, but rather a contraction on some neighborhood of a given point $y$, In this case we restrict our function to a certain open ball $B_{r}(y)$ so that $f : B_{r}(y)\rightarrow X$ is contraction map with contraction constant $h < 1$ and we assume that $d(y, f (y)) < r(1−h)$. Then, $f$ has a unique fixed point in the ball. Is this fixed point global? or does the fixed point change if you change the ball? and is this ball also unique?