An abjuration wizard has the Arcane Ward class feature (PHB, p. 115):
Starting at 2nd level, you can weave magic around yourself for protection. When you cast an abjuration spell of 1st level or higher, you can simultaneously use a strand of the spell's magic to create a magical ward on yourself that lasts until you finish a long rest. The ward has hit points equal to twice your wizard level + your Intelligence modifier. Whenever you take damage, the ward takes the damage instead. If this damage reduces the ward to 0 hit points, you take any remaining damage.
If an abjurer is dropped to 0 hit points by means other than damage (such as by failing a saving throw against a Banshee's Wail), their Arcane Ward remains active (assuming it was active before). What happens if they are attacked whilst dying?
- They take a step towards death; the rules on Damage at 0 hit points don't take the Arcane Ward into account;
- The Arcane Ward continues to "block" damage, so the ward would need to be reduce to 0 hit points/the damage exceed the ward's hit point before it counts as the wizard taking damage at 0 hit points. Hence if the ward had 10 hit points and a monster dealt 5 hit points, the wizard would not suffer any failed death saving throws;
- Both of the above; the Arcane Ward would continue to absorb damage, which may be relevant for death by massive damage, but the wizard would still suffer failed saving throws anyway);