Earliest work showing an Empire of Mankind

There is a literal empire in The Struggle for Empire (1900) by Robert William Cole. In name it is the "Anglo-Saxon Empire," but as the story explains in the introduction it encompasses all of the people of Earth (the world having been largely conquered by the U.K., U.S. and Germany who merged into the Anglo-Saxon Empire) and, by subsequent expansion, the rest of the solar system.

Once it had expanded to all the planets and major moons of the solar system (all of which are habitable in this universe) it started sending expeditions to the stars. Much of the exploration/exploitation is privately funded, and takes place off-screen:

The eager Anglo-Saxons turned their attention to other fields of acquisition, and plunged further and further into the depths of space, some even trying, in the pride of their power and their thirst for science, to seek God on His throne in those unknown regions. Dim accounts had been handed down from generation to generation of a certain great man named Napoleon Bonaparte who once nearly conquered the world. Now there were thousands of Bonapartes endowed with colossal intellect, vast energy, and boundless ambition, each burning to wrest for himself a world from the great Unknown. Provinces and countries were not even thought of; they desired to rule over a planet, a system, a universe. There was present everywhere an intense fever for acquisition; men burned with a desire to plunder in these new regions.

_{The Struggle for Empire, Chapter I: Introduction}

The remote colony world of Iosia, however, does figure large in the story:

For a long time past, a band of adventurers had been colonizing a planet called Iosia, which was situated far beyond the confines of the Sirian System. This planet was particularly rich in mineral wealth and other natural products, hence its possession was considered by the Colonial Bureau to be a most valuable acquisition. But shortly after the first party of Anglo-Saxons had landed there, its existence became known to the people of Kairet, the ruling planet of the Sirian System, and they much coveted the enormous resources which it was rumoured to possess.

_{The Struggle for Empire, Chapter III: The First Note of War}

Note that there is an official "Colonial Bureau" which would only be necessitated by large-scale colonial activities. The planets of the Solar system are considered "federated worlds" and don't count as colonies; only worlds beyond are part of the colonial empire. ("Sirian System" refers to the planetary system of the "fixed star Sirius.")

After the defeat of the Sirians, the Anglo-Saxon Empire comes to encompass at least some of their worlds as well:

The Anglo-Saxons demanded the evacuation of the planet that had been the original cause of the war, the surrender of another planet called Maikat, that belonged to the Sirian system, the payment of a huge war indemnity, and the limitation of the Sirian war-ships for the future to their own planetary system.

_{The Struggle for Empire, Chapter XV: The End of the War}

So here we have a proper empire, with member worlds and occupied/exploited subject worlds, encompassing all of latter-day humanity; which I believe satisfies the question.