The sample project on github due to @zoul's answer is the way to go. At the time of this writing, that project does not actually put anything on the clipboard (pasteboard). here is how:

Change @zoul's implementation of this method to:

- (void) copy:(id)sender {
    UIPasteboard *pboard = [UIPasteboard generalPasteboard];
    pboard.string = self.text;  
}

In Swift 5.0 and Xcode 10.2

Add copy option to your UILabel directly in your ViewController.

//This is your UILabel
@IBOutlet weak var lbl: UILabel!

//In your viewDidLoad()
self.lbl.isUserInteractionEnabled = true
let longPress = UILongPressGestureRecognizer.init(target: self, action: #selector((longPressFunctin(_:))))
self.lbl.addGestureRecognizer(longPress)

//Write these all functions outside the viewDidLoad()
@objc func longPressFunctin(_ gestureRecognizer: UILongPressGestureRecognizer) {
    lbl.becomeFirstResponder()
    let menu = UIMenuController.shared
    if !menu.isMenuVisible {
        menu.setTargetRect(CGRect(x: self.lbl.center.x, y: self.lbl.center.y, width: 0.0, height: 0.0), in: view)
        menu.setMenuVisible(true, animated: true)
    }
}

override func copy(_ sender: Any?) {
    let board = UIPasteboard.general
    board.string = lbl.text
}

override var canBecomeFirstResponder: Bool {
    return true
}

override func canPerformAction(_ action: Selector, withSender sender: Any?) -> Bool {
    return action == #selector(copy(_:))
}

If you want to generalize this further to work for more than one label in your view controller, including labels that are buried deep in the view hierarchy, do this:

Add a variable to your view controller

private var currentLabelToCopy : UILabel? = nil

For each label that you want to add the copy function to, do basically what the above answer says to do in viewDidLoad. Here's an example for a label inside a tableView cell:

cell?.detailTextLabel?.isUserInteractionEnabled = true
let longPress = UILongPressGestureRecognizer.init(target: self, action: #selector((longPressFunctin(_:))))
cell?.detailTextLabel?.addGestureRecognizer(longPress)

Now modify the functions shown above as follows

@objc func longPressFunctin(_ gestureRecognizer: UILongPressGestureRecognizer) {
    if let lbl = gestureRecognizer.view as? UILabel
    {
        lbl.becomeFirstResponder()
        let menu = UIMenuController.shared
        if !menu.isMenuVisible {
            let frame = lbl.frame
            if let superview = lbl.superview
            {
                menu.setTargetRect(CGRect(x: frame.maxX - (lbl.width / 2), y: frame.maxY - (lbl.height / 2) - 5, width: 0.0, height: 0.0), in: superview)
                menu.setMenuVisible(true, animated: true)
                self.currentLabelToCopy = lbl
            }
        }
    }
}

override func copy(_ sender: Any?) {
    let board = UIPasteboard.general
    if let lbl = self.currentLabelToCopy
    {
        board.string = lbl.text
    }
}

These stay the same:

override var canBecomeFirstResponder: Bool {
    return true
}

override func canPerformAction(_ action: Selector, withSender sender: Any?) -> Bool {
    return action == #selector(copy(_:))
}

Also see this if you have any problems: https://stackoverflow.com/a/23839272/826946

Override the UITextField instance's textFieldShouldBeginEditing method, and set it to return NO in order to disable editing.

Take a look at the UITextFieldDelegate protocol for more details.

Swift 5.3 and SwiftUI

To make this work in SwiftUI we can use the method that pableiros created an combine that with a UIViewRepresentable.

There are two updates that we need to make to the CopyableLabel class as the following methods were deprecated in iOS 13.

.setTargetRect(_,in:)

.setMenutVisible(_,animated)

We can easily fix this by using the .showMenu(from:rect:) method instead.

Here is the updated CopyableLabel class.

class CopyableLabel: UILabel {

    override init(frame: CGRect) {
        super.init(frame: frame)
        self.sharedInit()
    }

    required init?(coder aDecoder: NSCoder) {
        super.init(coder: aDecoder)
        self.sharedInit()
    }

    func sharedInit() {
        self.isUserInteractionEnabled = true
        self.addGestureRecognizer(UILongPressGestureRecognizer(target: self, action: #selector(self.showMenu)))
    }

    @objc func showMenu(sender: AnyObject?) {
        self.becomeFirstResponder()

        let menu = UIMenuController.shared

        if !menu.isMenuVisible {
            menu.showMenu(from: self, rect: self.bounds) // <-  we update the deprecated methods here
        }
    }

    override func copy(_ sender: Any?) {
        let board = UIPasteboard.general

        board.string = text

        let menu = UIMenuController.shared

        menu.showMenu(from: self, rect: self.bounds) // <- we update the deprecated methods here
    }

    override var canBecomeFirstResponder: Bool {
        return true
    }

    override func canPerformAction(_ action: Selector, withSender sender: Any?) -> Bool {
        return action == #selector(UIResponderStandardEditActions.copy)
    }
}

Then to get this class to work with SwiftUI all we have to do is create a simple UIViewRepresentable.

struct CopyableLabelView: UIViewRepresentable {

    let text: String
    private let label = CopyableLabel(frame: .zero)

    init(text: String) {
        self.text = text
    }

    func makeUIView(context: Context) -> UILabel {
        // Set the text for the label
        label.text = text

        // Set the content hugging priority so the UILabel's view is
        // kept tight to the text.
        label.setContentHuggingPriority(.required, for: .horizontal)
        label.setContentHuggingPriority(.required, for: .vertical)
        return label
    }

    func updateUIView(_ uiView: UILabel, context: Context) {
        // Handle when the text that is passed changes
        uiView.text = text
    }
}
  

If you have multiline text, you should use UITextView

Set the delegate:

func textView(_ textView: UITextView,
              shouldChangeTextIn range: NSRange,
              replacementText text: String) -> Bool {
    return false
}

And it should work magically :)

2019 ...

Save anyone typing:

public class SomeComplexCustomView: UIView {

    @IBOutlet var oneOfYourLabels: UILabel!
    ... your other labels, boxes, etc

    public func makeThatLabelCopyable() {
        oneOfYourLabels.isUserInteractionEnabled = true
        addGestureRecognizer(UITapGestureRecognizer(
          target: self, action: #selector(self.copyMenu(sender:))))
        addGestureRecognizer(UILongPressGestureRecognizer(
          target: self, action: #selector(self.copyMenu(sender:))))

        // or use oneOfYourLabels.addGesture... to touch just on that item 
    }

    public override var canBecomeFirstResponder: Bool { return true }

    @objc func copyMenu(sender: Any?) {
        becomeFirstResponder()
        UIMenuController.shared.setTargetRect(bounds, in: self)
        // or any exact point you want the pointy box pointing to
        UIMenuController.shared.setMenuVisible(true, animated: true)
    }

    override public func copy(_ sender: Any?) {
        UIPasteboard.general.string = oneOfYourLabels.text
        // or any exact text you wish
        UIMenuController.shared.setMenuVisible(false, animated: true)
    }

    override public func canPerformAction(
      _ action: Selector, withSender sender: Any?) -> Bool {
        return (action == #selector(copy(_:)))
    }
}

It's that easy!

One subtlety:

One detail for better engineering:

Notice we turn on first responder:

 public override var canBecomeFirstResponder: Bool { return true }

Often, on a given screen with such a label, you either will or won't have a copyable link like this.

So you'll very likely have something like:

var linkTurnedOnCurrently: Bool = false

func doShowThatLink( blah ) {
    linkAvailableOnThisScreen = true
    ... the various code above ...
}

func doShowThatLink( blah ) {
    linkAvailableOnThisScreen = false
    ... perhaps de-color the link, etc ...
}

Thus, in fact instead of this:

 public override var canBecomeFirstResponder: Bool { return true }

be sure to do this:

 public override var canBecomeFirstResponder: Bool {
    if linkTurnedOnCurrently { return true }
    return super.canBecomeFirstResponder
 }

(Note that it is not something like "return linkTurnedOnCurrently".)