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 (in the special case of $k$ a non-archimedean local field) The map $G(k)/H(k)\to (G/H)(k)$ is a homeomorphism onto its image, always has closed image and has an open image when $H$ is smooth.

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 $0$).

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 (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 $0$).