Nothing built-in to the stat block
There isn't anything inherently in the Lich's *(MM, 325) stat block that gives them a non-magical attack.
But they aren't without options. The DM can give them a melee weapon (or they can take an unarmed strike - although that's just for 1 damage so that's not optimal by any means) if they really want it, but none of that does much damage and is worth their action.
The easiest 'counter' to be within an Anti-magic field is take the opportunity attack as they move out of it and then they can use any of the options they have (which a you've noted are all magical.) Once out, they can use their regular fully array of actions to deal damage, get away, end the anti-magic field or anything else that gives value to the Lich.
Lair actions are also off the table as a means to get around the AMF(my emphasis):
the lich can take a lair action to cause one of the following magical effects...
This is not a job for the lich
As user TimothyAWiseman wisely points out, this is not the best of the Lich. The Lich is highly unlikely to be alone. Minions may abound and be used to help protect the Lich from a caster getting close enough, or dealing with the caster to try and break their concentration. This is the perfect opportunity for those minions to come in and protect their master/mistress.