ICRC officials said a group of its vehicles were fired on Tuesday, leaving two workers injured and one missing. The missing worker is feared to be seriously injured but officials said the situation on the streets made a search impossible.
The Red Cross said in a press release that colleagues of the missing worker had attempted a rescue but had to "turn back because of the ongoing fighting."
The ICRC said its vehicles were clearly marked and bore flags identifying them as Red Cross vehicles.
"The exact circumstances of the incident are not known and it has been impossible to establish whether the ICRC team was caught in a crossfire or came under direct attack... Given the chaotic and totally unpredictable situation in the city, getting from one place to another involves incalculable risks," the statement said.
Reporters observed and military officials acknowledged widespread looting and chaos in Baghdad on Wednesday as Saddam Hussein's regime seemed to have lost its grip on power.
Fighting between U.S. forces and pockets of resistance has made it impossible to tend to the wounded lying in the streets an ICRC spokesman told CNN.
"Casualties have been seen on the roads, on some bridges, and there was no possibility of evacuating them, for the reason that there was immediate fire as soon as anybody was trying to approach," said Roland Huguenin-Benjamin from Baghdad.
The Red Cross was one of a few remaining international relief organizations operating in the city. It called on belligerents in the conflict to "grant safe passage to all vehicles and staff working under the red cross and red crescent emblems and to allow them to perform their live-saving tasks."
On Tuesday, the World Health Organization and the International Red Cross said Baghdad-area hospitals were overrun by wounded civilians and running low on key medical supplies.
"Before the war started, there were enough supplies for a normal medical situation," WHO spokesman Iain Simpson said. "This is not a normal situation."
Red Cross spokesperson Nada Doumani warned that the hospitals, facing as many as 100 casualties an hour during heavy fighting, are near collapse.
"The hospitals have reached their limit," Doumani said Tuesday. "Staff are working around the clock."
Doumani also said the doctors are running low on clean water, anesthetics and other critical supplies.