Let $X_{p} = \mathbb{Q}_{p} / \sim $, where $\sim$ is defined by:

$x\sim 0 \Leftrightarrow x\in \mathbb{Q}$

$X_{p}$ is path-connected, because (unless I'm making some horrible mistake,) for any $x\in X_{p} \backslash \mathbb{Q}$, we have that $\lbrace x,0\rbrace$ under the subspace topology is path-connected.

Is $X_p$ contractible?