Governor's Beach lies near Moni, and is easily accessible from the motorway. There are a variety of kiosks and taverns there, and sunbeds available for hire at the standard €2.50 fee. The beach itself is a lovely long stretch of sand that is kept very clean.
There are actually two parts to Governor's Beach. We just followed the signs and took the pictures when we hit the surf. This is what the internet tells us though:
_____Comprised of two parts (Governor’s and Kalymnos), the first section of Governor’s beach has dark grey sands and uneven surfaces, whereas the second section (Kalymnos) - which lies around 200 m west of Governor’s beach - is replaced by flat ground and a long, narrow stretch of sandy coastline.
_____Based on that, I would say we were at the Kalymnos end.
Lifeguards with lifesaving equipment and first aid services are stationed at the beach from April - October, 09:00 - 17:00, and until 18:00 during the summer season.