Can I obtain the user's IP address?

Yes.
You can obtain the IP address of the user who sent the API request by using {$smarty.server.REMOTE_ADDR} in a custom process linked to an endpoint.

For example, if you set it in the post-processing of an endpoint, you can use the following custom process to add the ip_address field to the response:

{append var='json' value=$smarty.server.REMOTE_ADDR index='ip_address'}
{assign var='processed_json' value=$json}