Metrics on $G/K$

Question

Let $G$ be a locally compact group and $K$ a maximal compact subgroup. Let $d$ be a metric distance on $G$ that is left-invariant. We may then assume it's $K$-right-invariant and if $G$ where also a Lie group then using the Lie algebra we can construct a left-invariant metric on $G/K$. In general, this does not seem to be possible. Are there any other significant or well-known cases when it is possible to define a left-invariant metric on $G/K$? Say, for $G(\mathbb{Q}_p)$ ($G$ an algebraic group)?