These flying mammals have been around for a while. Complete fossils
of them have been found that date to the Eocene period (54-38 million
years ago).
Icaronyctis was an Eocene bat. Fossilized remains of it have been
found in the Gree River formation of Wyoming. Paleontologists have
examinted fossils of Icaronyctis and found evidence that it may have
practiced echolation, i.e. using a built-in sonar to detect the
presence of things in the dark. Fossils of bats have been found in
many places, including the Messel Shale in Germany.
The oldest fossils of megachiropterans ("flying foxes" or "fruit
bats") have been unearthed in Italy. These fossils date to the
Oligocene period (38-26 million yeras ago). That is ironic, since
megachiropterans now do not inhabit Italy; they are found in tropical
areas (for example, in parts of Africa, Asia, and Australia). Some
megachiropteran fossils have been found in Africa. They date to the
Miocene (26-7 million years ago).

Neal Robbins

P.S. Some tentative evidence indicates that bats may have existed
during the Paleocene period (65-54 million years ago), which was the
first part of the Tertiary era. Small teeth resembling those of bats
have been found and dated to the Paleocene. However, no other bones
were found with them; hence it cannot be concluded that the teeth are
those of bats. Still, it is plausible that bats existed during the
Paleocene times.

P.P.S. The taxonomy of bats is:

Kingdom: Animalia
Phylum: Chordata
Class: Mammalia
Order: Chiroptera
Suborders: Megachiroptera and Microchiroptera
Megachiroptera has only 1 family; it is called Pteropodidae and has
166 species.
Microchiroptera has 16 families and 759 species.
The total number of bat species is 925.