I don't doubt the vastness of the British empire but I doubt the saying the sun never sets on the British empire was ever literally true. At point in history did the sun literally never set on the British empire?
While it was not true for the British Empire throughout its life, it certainly was true that the sun never set on the Empire at its height of its geographical extent in the 1920s.
If we run through the 24 time zones, we can see that each one was home to a British territory or mandate of some sort after the division of territories that ended World War I, and hence that while the sun was setting in one time zone, it was still daylight in the adjacent zone.
GMT - Great Britain
+1 - South West Africa (Namibia)
+2 - South Africa
+3 - Aden
+4 - Muscat & Oman
+5 - India
+6 - Burma
+7 - Singapore
+8 - Sarawak
+9 - Australia
+10 - Papua (New Guinea)
+11 - New Zealand
+12 - Baker Island
-11 - Niue
-10 - Gilbert & Ellice Islands
-9 - Canada
-8 - Pitcairn Island
-7 - Canada
-6 - British Honduras
-5 - Bahamas
-4 - Falkland Islands
-3 - British Guinea
-2 - South Georgia
-1 - Tristan da Cunha
Simply put, the answer is yes. Modern day Great Britain possesses territory dotted around the world, with one small island chain in the South Pacific called the Pitcairn islands, which joined the British empire in 1838 as a colony, acting as the only point of the British empire which sees sunlight from midnight GMT to 0100 GMT.
If you want to know the last time that the sun set on the lands ruled by the English Crown, it goes back pretty far. The Americas were colonized by England from 1607 onwards with Jamestown and various colonies in the Caribbean. This quarter of the world continues to have a British presence through the British Virgin Islands and the Falkland Islands.
In 1639, Great Britain created a colony in Madras, off the coast of India. This area continues to have a British presence through several small island states, such as the British Indian Ocean Territory. At this point, it was only a lack of a colony in the Pacific that allowed the sun to set on the British Empire.
This changed on 26 Janurary 1788 when a penal colony was established in Australia. As all the above colonies were less than 8 timezones apart, it could now be definitively stated that the sun never set on the Empire. Since the Pitcairn Islands are still part of the Great Britain's territories, this location in the world is covered.
While the original colonies in these necessary regions have generally been granted independence, nearby islands continue to cover their position to make the statement accurate, and have allowed it to be accurate for the past 229 years.
XKCD has an interesting bit on the modern day aspects of this saying, https://what-if.xkcd.com/48/