Is it possible to enable authentication in Kibana in order to restrict access to a dashboard to only be accessible to particular users?
Kibana itself doesn't support authentication or restricting access to dashboards.
You can restrict access to Kibana 4 using nginx as a proxy in front of Kibana as described here: https://serverfault.com/a/345244. Just set proxy_pass to port 5601 and disable this port on firewall for others. This will completly enable or disable Kibana.
Elastic also has a tool called Shield which enables you to manage security of elasticsearch. With Shield you can for example allow someone to analyze data in specific indexes with read-only permissions. https://www.elastic.co/products/shield
Edit: Elastic has an issue on github and they recommend to use Shield.
Remember Shield provides only index-level access control. That means user A will be able to see all dashboards but some of them will be empty (because he doesn't have access to all indices).