It is important to remove the bloodsuckers at all, since the parasites can transmit serious diseases. It usually takes several hours or days for a tick to transmit pathogens, it should get out immediately if it detects them.
If, with all due caution, a tick's head gets stuck, this is no reason to panic. Disinfect the area so that no bacteria can spread and just wait. If the inflammation does not subside and continues to spread, you should go to a veterinarian.