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

Swift has taken the approach where a sliced collection is simple a view into the storage of the original collection, rather than eagerly copying its elements into a new Array. The reason to have a specific ArraySlice type is so you are aware of the memory management around the contents of the original array – the slice is holding a strong reference to it, so it will stay alive even after other references have disappeared. This gives you control over when a copy occurs so you can optimise performance and memory usage.

This is explained further by Ben Cohen in What's New in Swift - WWDC 17 at 35 minutes in, which covers slicing strings but also applies to arrays and other collections.