An example would be a lich's Paralyzing Touch.

Since it’s innate, I would assume so, but I am not an expert here.

https://www.dndbeyond.com/sources/sac/sage-advice-compendium#SA224

 Ask yourself these questions about the feature:

• Is it a magic item?
• Is it a spell? Or does it let you create the effects of a spell that’s mentioned in its description?
• Is it a spell attack?
• Is it fueled by the use of spell slots?
• Does its description say it’s magical?

If your answer to any of those questions is yes, the feature is magical.

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So no it won't work. Since the Paralizing Touch is a "Spell Attack" it is magical and will not work in an antimagic field

No spells cannot pass. Only those from artifacts and cast by deities 

Quote from ACEspinz >>

https://www.dndbeyond.com/sources/sac/sage-advice-compendium#SA224

 Ask yourself these questions about the feature:

• Is it a magic item?
• Is it a spell? Or does it let you create the effects of a spell that’s mentioned in its description?
• Is it a spell attack?
• Is it fueled by the use of spell slots?
• Does its description say it’s magical?

If your answer to any of those questions is yes, the feature is magical.

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So no it won't work. Since the Paralizing Touch is a "Spell Attack" it is magical and will not work in an antimagic field

 Huh, alrightio then, nicely done!