Around here, I have not seen it done. That doesn't mean that installing a smoke detector in a lift car (elevator car) isn't required somewhere.
Codes and Rules
The requirements for smoke detectors are a gray area. What some people think of as "no-brainer to install" smoke detectors, other people think that smoke detectors are more of a "false alarm nuisance" than useful. Yes, I've had one fire marshal refer to smoke detectors as "no-brainer to install" and requires more smoke detectors; while 30 miles away, another fire marshal refers to smoke detectors as a "false alarm nuisance" and requires the smoke detectors to be removed.
In the United States, fire marshals can overrule the fire codes.
See Just Who Is the Authority Having Jurisdiction (AHJ)?
for what the NFPA says about the Authority Having Jurisdiction (AHJ).
It's important to know also, there is the possibility that something like smoke detectors in lift cars may have been required at one time, but now smoke detectors are forbidden in the rules and codes. In other words, at one time, smoke detectors may have been "Grandfathered" in the lift cars, but now with the fire marshal's permission, the may be removed.
To find out for sure, you probably need to talk to your local fire marshal.
Reason for Installing Smoke Detectors
In a building fire, smoke inhalation is the single greatest killer. Complete coverage of the whole building with smoke detectors may include installing the smoke detectors in the lift cars. This would be for the protection of the people who might be in the lift car.
Of course, it you're referring to a freight lift car (freight elevator car), there could be other concerns. Again, you probably need to talk to your local fire marshal.
Reason for Not Installing Smoke Detectors
70% or more fire alarms in buildings are caused by false alarms. Smoke detectors are more prone to going into false alarms than most other parts of a fire alarm system. If the smoke detectors are removed, there will be fewer false alarms. Then, when there is an alarm, people will take the alarm more seriously.
In a lift car, the wiring and maintenance is more difficult for fire alarm technicians. Two companies have to coordinate together (the fire alarm company and the lift company) to service the wiring and the detector.
Also, as they are bundled together for the distance of the lift's traveling cable, electrical crosstalk between the high-power wires and the low-power fire alarm wires means a greater chance of false alarms, or other issues. This especially becomes an issue when the fire alarm control system is changed in later upgrades, and the traveling cable is not changed.
Talk to the Fire Marshal
Talk to the fire marshal, the fire marshal is the government's representative and knows what should be done and what shouldn't be done according to the government.