Have header image there but not linking to homagepage

Hi there,

Try removing the logo in Site Identity and just add it as a static image using the before header content hook:
http://demo.generatepress.com/hook-locations/
https://docs.generatepress.com/article/hooks-overview/

If you need more info, can you provide a link to your site? Thanks!

Or you could filter the logo HTML (because filters are awesome):

add_filter( 'generate_logo_output','tu_no_logo_link', 10, 2 );
function tu_no_logo_link( $output, $logo ) {
	printf( 
		'<div class="site-logo">
			<img class="header-image" src="%2$s" alt="%1$s" title="%1$s" />
		</div>',
		esc_attr( apply_filters( 'generate_logo_title', get_bloginfo( 'name', 'display' ) ) ),
		esc_url( apply_filters( 'generate_logo', $logo ) )
	);
}