Can the free gift I send to members be a membership product?

If you set the free gift you send your members to a product associated with a membership, the order will still be added to your Shopify store with the product, however it will not actually create a new membership for the customer. This is because the order created doesn't trigger a new subscription contract to be created and associated with the customer. 

There is currently no way to send a "free gift membership" to your members.