Based on the information I have found, the Lightning Network implements a network fee, but this fee is charged only once when a channel (for transactions) is opened, and we are allowed to make any number of transactions while the channel is open. For more details, please see the following information: Lightning Network Explained: A Deep Dive into Bitcoin’s Scalability Solution

There is also a thread that collects discussions related to the Lightning Network, please try reading it: Bitcointalk.org @Pmalek - All Lightning Network Threads In One Place

As an additional point of reference, El Salvador ever adopted the Lightning Network. Then, it also ever been used to fund Robert F. Kennedy's presidential campaign, and this was the first time a presidential campaign used Bitcoin as the first reference said, but, It seems there is also other information that says that Donald Trump was the first (I don't really understand which one is correct) : Trump Campaign Adopts Bitcoin Lightning Network for Donations