I'm not ready to bet serious money on "first in the world": claims like that are often made without adequate footnoting! But the U.S., technologically backwards though it seems today in railroad matters, was at the leading edge at the beginning of the 20th C, so first in the U.S. might well have been the first in the world.
And first in the U.S. may well have been on a New York subway. Details on the relevant railroad, then a story about the cars.
The oldest part of the new York subway system is traditionally called the IRT, for the Interborough Rapid Transit company that built it. (Officially the nomenclature IRT/BMT/IND for the three components of the NY subway system has been dropped, but I'll bet most New Yorkers still use it. IRT lines are, for the most part, the ones with numbers on current subway maps-- BMT and IND lines have letter designations.) The first part of the IRT network to go into service opened in October 1904: the original line started in lower Manhattan, went up the East Side to 42nd Street, crossed to the West Side, then up to 145th Street: to ride the route today you'd need to ride three trains: East Side IRT (4, 5 or 6 train) from lower Manhattan to Grand Central, then the 42nd street Shuttle to Times Square, then the West Side (Broadway) IRT (number 1 local) up to Harlem. This route is (except for a viaduct crossing 125th Street in the "Harlem Valley") entirely underground, and safety and operational concerns motivated a search for new carbuilding technology.
The IRT published a book boasting about itself in 1904: "Interborough Rapid Transit: The New York Subway: its construction and equipment." This was reprinted some time around 1970 by Arno Press. (Weirdly, there doesn't seem to be a date in the reprint, but the Library of Congress Card Number is 71-90436, and the first two digits of LC numbers are APPROXIMATELY the year of publication-- typically off by one or two years, but I don't remember in which direction.)
According to this souvenir book, "At the time of placing the first contract for the rolling stock of the subway, the question of using an all-steel car was carefully considered by the management. Such a type of car, in many respects, presented desirable features for subway work as representing the ultimate of absolute incombustibility." Alas, this wasn't possible when the contract for the initial 500 cars was placed in 1902: "no cars of of this kind had ever been constructed." So the subway opened with cars of mixed (partly metal, but a fair bit of wood) construction.
Continuing, however, "The plan of an all-metal car... was not abandoned, and although none was in use in passenger service anywhere, steps were immediately taken to design a car of this type and conduct the necessary tests to determine whether it would be suitable for railway service. None of the car-building companies was willing to undertake the work, but the courteous coöperation of the Pennsylvania Railroad Company was secured in placing its manufacturing facilities at Altoona at the disposal of the [IRT company]."
(Comment: More than courtesy was involved on the PRR's part, of course. They were working on their access to New York City, which involves a significant bit of tunnel trackage -- Amtrak's line from New Jersey to Penn Station in NYC and on to Long Island -- and so had an interest in all-steel, "incombustible," cars: I think that right from the inauguration of service to Penn Station in NYC only all-steelcars were allowed in the PRR's Hudson River tunnels.)
A prototype car was completed in December 1903... and was too heavy. So redesign work ("involving very original features") was undertaken, and when the 1904 book was written an order for 200 all-steel cars was under construction.
So your co-worker may be right!
(And, for all that New Yorkers now take it for granted, the IRT subway represented major innovation in railroad technology.)