The problem is that mentions[0...3] returns an ArraySlice<String>, not an Array<String>. Therefore you could first use the Array(_:) initialiser in order to convert the slice into an array:

let first3Elements : [String] // An Array of up to the first 3 elements.
if mentions.count >= 3 {
    first3Elements = Array(mentions[0 ..< 3])
} else {
    first3Elements = mentions
}

Or if you want to use an ArraySlice (they are useful for intermediate computations, as they present a 'view' onto the original array, but are not designed for long term storage), you could subscript mentions with the full range of indices in your else:

let slice : ArraySlice<String> // An ArraySlice of up to the first 3 elements
if mentions.count >= 3 {
    slice = mentions[0 ..< 3]
} else {
    slice = mentions[mentions.indices] // in Swift 4: slice = mentions[...]
}

Although the simplest solution by far would be just to use the prefix(_:) method, which will return an ArraySlice of the first n elements, or a slice of the entire array if n exceeds the array count:

let slice = mentions.prefix(3) // ArraySlice of up to the first 3 elements

Just initialize a new Array with the slice:

let arr1 = [0,1,2,3,4,5,6,7]
let slice = arr1[2...5]
let arr2 = Array(slice)

Creating an ArraySlice from an array is cheap - O(1), so convert the original array into an ArraySlice for the first function call, and then deal just with ArraySlices.

You can also overload the method to accept both:

func functionName(arr: [Int]) -> Int {
   functionName(arr: arr[...])
}

func functionName(arr: ArraySlice<Int>) -> Int {
   // ...
}