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The unconscious state description says

Any attack that hits the creature is a critical hit if the attacker is within 5 feet of the creature.

Situation

Target creature is unconscious. Attacker stands in 5 feet near it. Attacker attack the target creature with savethrow-type attack of any kind. Frostbite cantrip for example.

Question

Does this attack cause critical hit?

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An attack is anything with an attack roll; saving throws do not have an attack roll so they are not attacks and do not interact with the unconscious rule.

The Player's Handbook states:

If there's ever any question whether something you're doing counts as an attack, the rule is simple: if you're making an attack roll, you're making an attack.

(See the question "What counts as an attack?" for further evidence/help)

Since the frostbite cantrip involves a saving throw and not an attack roll it is not an attack and is not turned into a critical hit. In fact, saving throws cannot be critical hits at all because they don't even hit. Attacks are the only things that can either hit or miss, and so they are the only things that can critically hit or miss as well. This is explained in the question "If you roll a 1 on a saving throw against a damage spell, do you take extra damage?"

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