The classical Busemann-Feller lemma in Euclidean space says the following.

Let $K\subset \mathbb{R}^n$ be a closed convex set.
Then

1. for any point $x\in \mathbb{R}^n$ there exists unique nearest point in $K$; let us denote it by $p(x)$.

2. the map $p\colon \mathbb{R}^n \to K$ does not increase distances, i.e. is 1-Lipschitz.

Question. Does this result hold in an $n$-dimensional hyperbolic space? A reference would be helpful.

Remark. I think part (1) is true for hyperbolic and spherical spaces; in the spherical space one should require in addition that $K$ is contained in an open half-sphere. I think part (2) is not true in the spherical space.