Drush site install without dropping/recreating database I am from Ukraine, so sorry for my bad English...
Please help me to fix a problem. I'm trying to install Drupal with Drush on shared hosting.

I use the command drush site-install. On localhost everything worked correctly, but on the  hosting I get an error: Access denied for user 'bigferwn_t'@'localhost' to database 'bigferwn_t'.

Command string is 100% correct. The problem lies with permissions. When Drush runs the site-install command it DROPs the RECREATEs the db. So if the db user does not have that permission (like in many hosting provider) we get the error access denied.

So I created a db in hosting account manually and want to install drupal using the database that I just created. I don't want to drop and recreate this db. I simply want use the existing database.

There are issues discussing the same problem https://www.drupal.org/node/1105514.

In the site-install command there is no option to "NOT CREATE OR DROP DB" or if there is I can't find it.

I tried to use the patch from https://www.drupal.org/files/issues/drush-script-options.patch which was proposed in the issue above, but it is old and for a different version of drush.
Drush doesn't and likely isn't going to support this workflow based on developer comments in these pull requests:

https://github.com/drush-ops/drush/issues/134

https://github.com/drush-ops/drush/pull/353

Your remaining option is to install using the Drupal installer script.