Bitbucket 7.15 introduced the “Branch deletion on merge” setting, available in Workflow > Branches on your project or repository. Auto-Merge will respect this setting when merging a pull request. This means that Auto-Merge will follow the behavior specified in that setting when merging a pull request.
If the version of your Bitbucket instance is 7.14 or lower, then Auto-Merge will delete the source branch on merge.