Although FSCalendar does not directly supports range selection, it does allow multiple selections, which means that at some point you would be able to handle the range selection by yourself.
So, in the viewDidLoad() you have to make sure that you set the calendar allowsMultipleSelection property to true:
private weak var calendar: FSCalendar!
override func viewDidLoad() {
    super.viewDidLoad()
    
    calendar.allowsMultipleSelection = true
}
The view controller should conform to FSCalendarDelegate protocol for handling the logic of selecting/deselecting a range.
What we need so far is to declare two dates for the range (the starting date and the ending date):
// first date in the range
private var firstDate: Date?
// last date in the range
private var lastDate: Date?
also an array of dates to hold value dates between firstDate and lastDate:
private var datesRange: [Date]?
and then implement the didSelect date and the didDeselect date delegate methods as:
func calendar(_ calendar: FSCalendar, didSelect date: Date, at monthPosition: FSCalendarMonthPosition) {
    // nothing selected:
    if firstDate == nil {
        firstDate = date
        datesRange = [firstDate!]
        
        print("datesRange contains: \(datesRange!)")
        
        return
    }
    
    // only first date is selected:
    if firstDate != nil && lastDate == nil {
        // handle the case of if the last date is less than the first date:
        if date <= firstDate! {
            calendar.deselect(firstDate!)
            firstDate = date
            datesRange = [firstDate!]
            
            print("datesRange contains: \(datesRange!)")
            
            return
        }
        
        let range = datesRange(from: firstDate!, to: date)
        lastDate = range.last
        
        for d in range {
            calendar.select(d)
        }
        
        datesRange = range
        
        print("datesRange contains: \(datesRange!)")
        
        return
    }
    
    // both are selected:
    if firstDate != nil && lastDate != nil {
        for d in calendar.selectedDates {
            calendar.deselect(d)
        }
        
        lastDate = nil
        firstDate = nil
        
        datesRange = []
        
        print("datesRange contains: \(datesRange!)")
    }
}
func calendar(_ calendar: FSCalendar, didDeselect date: Date, at monthPosition: FSCalendarMonthPosition) {
    // both are selected:
    
    // NOTE: the is a REDUANDENT CODE:
    if firstDate != nil && lastDate != nil {
        for d in calendar.selectedDates {
            calendar.deselect(d)
        }
        
        lastDate = nil
        firstDate = nil
        
        datesRange = []
        print("datesRange contains: \(datesRange!)")
    }
}
What about datesRange method?
I did not mention it in my answer for the purpose of making it shorter, all you have to do is to copy-paste it from this answer.
Output:
