2024 Lich and Anti-Magic Field, how boned is the bone man?

I've got some players wanting to find a way to have anti-magic field on my campaigns BBEG and I was curious what the ruling is on what part of the Lich's stat block is affected. 

Obviously none of the spells would be cast while in the AMF, and his legendary action that casts fear would be a no go as well. But both of the Lich's attacks are listed as melee (or ranged) attacks does that mean they can be used while in the AMF field since they aren't listed as being magic attacks or spells? 

Or since basically everything a Lich can do is magical in nature, does that mean attaks are only going to be unarmed attacks?

Anti magic field totally screws with Wizards and similar creatures.  The main problem with the spell is that it can only be cast by Wizards and Clerics.

I know of few Wizards that take the spell because it screws them as much as it screws their enemy.  But armored tank clerics like the spell. 

Any Lich will know that they need to counterspell the AMF if they have the chance. Once it's up, the Lich can move to stay away from the caster. Depending on how paranoid the Lich is, they might have physical means to deal damage to the caster, such as traps and minions. If they are particularly arrogant, they might have thought that this can't have been enough to stop them, so it could be a good hard counter!

A Lich is an intelligent creature and an undead immortal to boot, so unless they're in the middle of a very time-sensitive spell or event, they can move away, teleport away, and then come back in an hour to continue the fight. Frustrating as this might be to the players, the Lich is not there to give them a good bit of sport - it is there to destroy them (likely in self-defence!)