Can I safely plug something that draws 6 amps into the cigar lighter (power port) in my Piper Archer lll?

Cigar lighters usually have a 10amp breaker and wiring to match, but the sockets themselves are pretty poor quality and only designed to be active for 30 seconds or so. They can get pretty hot when used for any length of time, such as an air-bed inflator or tyre pump, even when the load is well below the breaker's limit.

If your use is at all critical, I'd be tempted to replace it with a bulkhead-mounted XT60 or similar connector.

Yes, you can safely plug in a 6A load. Possibly a circuit breaker will pop out after some time. That depends on the rating of the circuit breaker serving the outlet and if there are any other loads on that circuit breaker. But it is completely safe to try it.