Date: 201911
Local time of day: 1201-1800
Location: ZZZ.ARTCC, US
Flight conditions: Marginal
Weather elements / visibility: Cloudy
Light: Daylight
Aircraft: Light Transport
Operator: Fractional
Operating under: Part 135
Flight plan: IFR
Mission: Passenger
Flight phase: Climb
Airspace: Class E ZZZ
ATC / advisory: Center ZZZ
Second aircraft: Any Unknown or Unlisted Aircraft Manufacturer
Reporter organization: Fractional
Reporter function: Captain; Pilot Not Flying
Reporter qualification: Flight Crew Instrument; Flight Crew Air Transport Pilot (ATP); Flight Crew Multiengine
Human factors: Confusion; Communication Breakdown
Second reporter function: First Officer; Pilot Flying
Anomaly: ATC Issue All Types; Conflict NMAC; Deviation - Speed All Types; Deviation / Discrepancy - Procedural Published Material / Policy
Detector: Person Flight Crew
Result: Flight Crew Took Evasive Action
Contributing factors: Human Factors
Primary problem: Human Factors

Narrative: Before departure from ZZZ; we picked up a clearance and got a hold for release. We contacted Center when we were number one for takeoff Runway XX. Center told us we were released; release time was XA40; void if not off by XA50; contact them no later than XA55 if not off. At XA42 we lifted off Runway XX and heard a plane in a panicking voice saying; Aircraft X; Center says 'DON'T TAKE OFF DON'T TAKE OFF.' We immediately contacted Center and he told us to make an immediate turn to 360; we were passing through 3;000 and he told us to descend to 3000 ft. We could see a plane closing in on the TCAS and banked about 45 degrees and pitched the nose down to start descent to 3;000. At that point we exceeded the flap speed. I immediately pulled flaps up. We leveled at 3;000 on a 360 degree heading. I asked the Controller where that traffic came from and he said; 'That was my fault. I was thinking you guys were coming off of ZZZ1.' As we were in the descending turn taking evasive action to avoid the traffic; I didn't get flaps up in time and we had a Flap Overspeed. Aircraft yelling don't take off when we were on climb out. Controller mistook our departure airport and released us for departure while there was traffic conflicting with our departure path and clearance. Steep descending turn to avoid traffic. Don't have any suggestions at this time. It was an honest mistake and glad it wasn't worse than it could have been.


Second narrative: Prior to departure; the Captain received our clearance from ATC and we were advised hold for release. We contacted ATC at the runway and advised we were number one and ready for takeoff. ATC released us for departure with a release void time of XA40; if not off by XA50; contact ATC no later than XA55. The Captain and I completed the before takeoff checklist and we began our departure roll down the runway. As we were lifting off the runway; another aircraft in the airport environment began broadcasting on the Unicom frequency looking for an 'Aircraft X.' The other aircraft called out again for an Aircraft X and said 'Center says don't take off.' [The Captain] immediately switched to the departure frequency and contacted Center. ATC Controller advised an immediate left turn to a heading of 360 and descend to 3;000. As [the Captain] called for me to turn and descend; I was already in a left descending steep turn to a heading of 360 and 3;000 ft. While during this event; [the Captain] was communicating with ATC; the aircraft accelerated past the flap speed; I immediately reduced the throttles to idle in order to correct my speed. When [the Captain] asked the Controller where the traffic came from; the Controller replied that it was his fault; he thought we were departing from ZZZ1. Aircraft in the airport environment during takeoff calling out 'don't take off.' ATC Controller mistook our departure airport for a different airport in the area. Steep descending turn to avoid traffic. No suggestion.


Synopsis: Light Transport aircraft flight crew reported departing from an uncontrolled airport that resulted in an NMAC due to ATC believing the flight had departed from a different airport.