From the number of votes, it seems that the useful comment of Laurent Moret-Bailly has been under appreciated, so I thought it would be useful to explicitly record what the main theorem in the linked paper says (in community wiki mode, since this is really his answer).

**Theorem 1.2** of [GGMB14][1] (in the special case of $k$ a non-archimedean local field) The map $G(k)/H(k)\to (G/H)(k)$ 

$\bullet$ is always a homeomorphism onto its image,

$\bullet$ has closed image if $H$ satisfies the condition ($\ast$),

$\bullet$ has an open image if $H$ is smooth.

The condition ($\ast$) appearing in this theorem is technical (see Definition 2.4.3 of the paper), but $H$ satisfies the condition ($\ast$) if it has either one of the following properties: smooth, unipotent, commutative, or being a normal subgroup of a smooth group. Interestingly, there are examples of $H$ not satisfying $(\ast)$ for which the map $G(k)/H(k)\to (G/H)(k)$ has non-closed image (see example 7.1 of the paper, taking for example $k=\mathbf{F}_p(\!(T)\!)$).

Note that in characteristic $0$, (affine) group schemes (of finite type) are always smooth by a result of Cartier. Also, this puts in perspective the comment of YCor on the non-oppeness of $\text{SL}_p(K)\to \text{PGL}_p(K)$ for $K = \mathbf{F}_p(\!(T)\!)$. Finally, let me remark that the implicit function theorem should prove in all characteristic the openness of $G(k)/H(k)\to (G/H)(k)$ when $H$ is smooth (as suggested in Venkataramana's answer in characteristic $0$).



[1]: http://content.algebraicgeometry.nl/2014-5/2014-5-025.pdf