Invoke AWS Lambda and return response to API Gateway asyncronously

With API Gateway's "Lambda Function" integration type, you can't do this with a single Lambda function -- that interface is specifically designed to be synchronous. The workaround, if you want to use the Lambda Function integration type is for the synchronous Lambda function, invoked by the gateway, to invoke a second, asynchronous, Lambda function through the Lambda API.

However, asynchronous invocations are possible without the workaround, using an AWS Service Proxy integration instead of a Lambda Function integration.

If your API makes only synchronous calls to Lambda functions in the back end, you should use the Lambda Function integration type. [...]

If your API makes asynchronous calls to Lambda functions, you must use the AWS Service Proxy integration type described in this section. The instructions apply to requests for synchronous Lambda function invocations as well. For the asynchronous invocation, you must explicitly add the X-Amz-Invocation-Type:Event header to the integration request.

http://docs.aws.amazon.com/apigateway/latest/developerguide/integrating-api-with-aws-services-lambda.html

Yes, simply create two Lambda functions. The first Lambda function will be called by the API Gateway and will simply invoke the second Lambda function and then immediately return successfully so that the API Gateway can respond with an HTTP 200 to the client. The second Lambda function will then take as long as long as it needs to complete.