SEOUL—Samsung Electronics Co. said on Tuesday that it is discontinuing its embattled Galaxy Note 7 smartphone, pulling the plug on a product whose botched recall has brought headaches to consumers and damaged the Samsung brand.

Samsung said in a filing with South Korean regulators that it would permanently cease production and sales of the device, following a string of reported incidents in which supposedly safe replacements of the smartphone overheated and in some cases caught fire.

“Taking...