Using Twig how can I get the summary of the body field to use in a template?
The only way I managed to solve this is:

if you only want to target the value of the summary, you can target the entity you're on. If you're on a node then you can do this :

node.body.summary


In a paragraph

paragraph.field_body.summary


You can also get the summary front content like this :

content.body['#object'].body.summary


Not great but will do for now.