Calling AmazonS3Client.GetObjectAsync() on lambda function take too long (~5s) on the first invoke

Question

I'm using S3 client in lambda function, this is my code:

public class SomeLambdaFunction
{
    public async Task<string> Handler(string input, ILambdaContext context)
    {
        var client = new AmazonS3Client();
        // begin stop watch
        await client.GetObjectAsync("<bucket-name>", "<object-key>");
        //end stopwatch and see took ~5s
    }
}

when I deploy and invoke in the first time, it took ~5s to execute client.GetObjectAsync, when I run later (reuse lambda instance), it is complete instantly.

PS: I'm aware of cold start issue of lambda function, but I thought it happen before lambda's handler actually run, does cold start really happen inside lambda's handler?