So, if I understand well, you want to use a specific page as search results page.
This is a bit tricky, because when you are on search page, current query is the one created by the search term in the url, but to get a specific page you need a query for that page.
So you need 2 queries. This is not ideal about performance, and probably using a different way to link banner to pages you can solve this problem in a more efficent way.
However, let's answer you question.
When you are in search result page WordPress uses the
search.php template to show the results. See WordPress template hierarchy.
search.php is not there, your theme is using
index.php to show search results, in this case duplicate the
index.php renaming the copy
search.php: in this way you have a specific template for search results.
If you are using a theme developed by third party, is a good idea doing everything in a child theme, in this way when you update theme you'll not lose the changes.
You don't say how you are linking banner to pages, I assume you upload banners as featured images, if you do it differently (e.g. using custom fields), edit the code accordingly.
search.php and where you want to show the banner use:
$post = get_page_by_path('search');
// now the page with slug 'search' is treated as it was the current page
// so retrieve the banner just like you do in normal page
// I'll use page featured image
Now, you have to go in your dashboard, create a page and assign it the slug 'search'. That slug is assigned automatically if you use 'Search' as title.
Finally add the banner to that page.