Continue screen recording even after screen turns off
No
There are no options for screen recording in the settings app. You also get a little notification that the recording has been stopped after you set the device to sleep.

If you own a Mac you could use QuickTime player to record the screen. Connect your device to the Mac with a cable then open QuickTime Player, right click on the icon in dock, select new recording and set the input to your mobile device (small arrow next to the red record-button). Then click record. It will continue the recording even though you set the mobile device to sleep.