A number of people were trapped inside the building as shots were exchanged.
The motive remains unclear. The Planned Parenthood group has drawn anti-abortion protests in the past.
A law enforcement source identified the suspect as Robert Lewis Dear, from North Carolina. No other details were given.
"I want to convey to the loved ones of the victims, this is a terrible, terrible tragedy that occurred here in Colorado Springs today," Mayor John Suthers told a news conference.
"Obviously, we lost two civilian victims. We mourn the loss of a very brave police officer." The dead policeman was named as Garrett Swasey, 44, who was married with two children.
Colorado Springs Police Chief Peter Carey said five police officers were among the injured, who were being treated in local hospitals.
Police had sealed off streets around the centre as officers tried to make contact with the suspect. "We did get officers inside the building," police Lt Catherine Buckley said.
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