Yes, current service status is available at status.haveibeenpwned.com. This page shows the operational status of the website, API, notification emails, and domain search.
If you're experiencing an issue with the API or dashboard, check the status page first to see whether there is a known incident.
Note that HIBP does not offer formal SLAs or uptime guarantees. See Do you have any published service level agreements (SLA) or uptime guarantees? for more detail. Issues are resolved as quickly as possible when they arise.