How to delay a CABasicAnimation?

Shouldn't you be using the [CAMediaTiming beginTime] property (reference)?

See Customizing the Timing of an Animation in the Core Animation Programming Guide.

CABasicAnimation *animation;
animation.beginTime = CACurrentMediaTime() + 0.3; //0.3 seconds delay

In Swift 3.0:

func animateYourView () {
   let myDelay = 5.0
   let scalePulseAnimation: CABasicAnimation = CABasicAnimation(keyPath: "transform.scale")
   scalePulseAnimation.beginTime = CACurrentMediaTime() + myDelay
   scalePulseAnimation.duration = 0.5
   scalePulseAnimation.repeatCount = 2.0
   scalePulseAnimation.autoreverses = true
   scalePulseAnimation.fromValue = 1.0
   scalePulseAnimation.toValue = 0.5
   myView.layer.add(scalePulseAnimation, forKey: "scale")
}

Where the key line for the delay is:

  scalePulseAnimation.beginTime = CACurrentMediaTime() + myDelay