The victim, a woman in her 60s, was using the crossing in Lower Rushall Street just after 10am on Tuesday when she was hit by the VW Polo.
West Midlands Police said the car drove on before stopping a short distance away.
A suspect was then held by members of the public assisted until officers arrived, the force said.
The cyclist was treated by paramedics, but was pronounced dead at the scene.
A man, aged 42, has been arrested on suspicion of causing death by dangerous driving and driving while unfit through drink or drugs. He remained in police custody on Tuesday afternoon for further questioning.
Officers have spoken with a number of witnesses at the scene and are retrieving CCTV footage as the investigation into the crash continues.
Detective Sergeant Julie Lyman, of the Serious Collision Investigation Unit, said: "We would like to thank members of the public for their assistance following this tragic incident.
"We are continuing to gather information about the precise circumstances of the collision and would urge anyone who can help, especially if they have dash-cam footage, to get in contact with us.”
Officers from West Midlands Police's Serious Collision Investigation Unit have been combing the scene, including looking over the VW Polo. They were seen performing skid tests, accelerating then quickly braking inside the cordon.
Paramedics from West Midlands Ambulance Service, as well as the air ambulance, were also called to the scene.
One witness, who wished to remain anonymous, said he thought he heard a "street sweeper" but went outside to find an air ambulance trying to land, before seeing CPR being performed on the cyclist.
He added: "It is shocking to be honest. You don't expect anything like this to happen."
A West Midlands Ambulance Service spokesman said staff "immediately began administering advanced life support" after finding the woman with serious injuries.
“However, despite their best efforts it sadly became apparent nothing could be done to save the woman and she was confirmed dead at the scene," he added.
Anyone with information or dash-cam footage is asked to message West Midlands Police via live chat at west-midlands.police.uk or call 101, quoting log 1195 of 13 July, or you can email FL_COLLISION_INVEST@west-midlands.pnn.police.uk