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!)

I'd see an antimagic field as a tool of defense rather than attack, against a lich.  Especially so, if the lich can fly or use Spider Climb, and the caster didn't think to bring those spells.

The lich stat block (if I'm looking at the same one as you) doesn't seem to include any kind of "antimagic susceptibility" trait like you'd see on Animated Armor or a Flying Sword.  Based on that, my thinking would be that the lich itself, or the effect(s) binding its soul to its body in the material plane, is something that the antimagic field doesn't affect, like something "created by an artifact" (like the lich's phylactery?).  So with that in mind, there might be some room to rule that a few of the lich's abilities are an inherent part of that effect.  So while the lich wouldn't be able to cast spells, summon creatures, or use magic items within the field, they might still have a couple of surprises for the party. 

Plus there's always the possibility that the lich possesses an artifact, which isn't affected by the field.  On one hand, that might feel unfair if it's not foreshadowed.  On the other hand, it's a lich, so there really shouldn't be "this one simple trick" to dealing with them.  No matter how prepared you think you are, the lich has potentially had lifetimes to anticipate it, and prepare countermeasures.