At first glance, I thought this was a simple question. But there’s more to it. So…
First you have to know that Obsidian supports two different kinds of links for internal linking:
- Markdown links, typically in the form
[cat](birthday.md) - Wiki style links, typically in the form
[[birthday]]or[[birthday|cat]]
The same applies for embedding local images:
- Markdown image embed, typically in the form
 - Wiki style image links, typically in the form
![[cat.png]]
If you want an image as a link, you need to place an image embed inside a link. However, there are only some combinations that work and some that don’t.
Placing a wiki embed in a Markdown link does work:
Placing a Markdown embed in a Markdown link should work, but on my side the link doesn’t work as expected. (Might be related to this bug in Obsidian.)
[](birthday.md)
Placing Wiki or Markdown embeds in Wiki style links seems not to work either. (I haven’t checked if there’s a feature request.)
[[birthday|![[cat.png]]]]
[[birthday|]]