I am just working on my harmony 9, and I am wondering if you can have non-chord tones in the bass?
Depending on where it is applied and how, and disregarding passing tones and the like, what could be non-chord tones in the bass actually end up changing the harmony. For a simple example, if the chord on the chart/being played by others is a G chord and the bass player plays an E beneath it, the chord becomes an E-7. Once you start adding different notes beneath a chord, it starts to get a bit more complicated trying to analyze the chord, so people often use slash chords instead of trying to give them a name of their own. This is particularly useful outside of Jazz, where Jazz has an extensive set of chord types as part of the standard repertoire and most other genres don't often use them. For instance, if you were to play Emin/F in a Jazz context, it would easily be reinterpreted as a sort of Fmaj7 #11, where in a pop context, most players wouldn't really know what to call it/how to think of it other than Emin/F.
Getting to the passing tones and neighbor tones, etc., these are incredibly common and part of most bass players' vocabulary, which I suspect is not what you're asking about.
Leaving out passing tones, the answer is, as so often, yes and no. And it's very much up to how you choose to analyse it. Does a 'slash chord' define a new chord or the basic one with a 'non-chord' bass note? Is 'C/Bb' a different chord to 'C7/Bb'? What about 'C/F#', the famous final chord of the show 'West Side Story'?
But I think that's too complicated an answer for the intention of this question. Yes, there are 'non-chord' bass notes. When we write F/G, the G note is used ONLY in the bass, it isn't added to the upper structure.
As passing tones, absolutely. For example if you're going from an F7 to a G7, an F# (which isn't in either chord) works very well to transition. For other notes, sometimes you'll see tones written for the bass that normally wouldn't be a part of the chord (but since they're written in I guess they're technically now chord tones?) For example, the progression "Stairway to Heaven" Am7, Am7/G#, Am7/G... The second cord is an Am7 (A-C-E-G) played with a G# in the bass.
Unless you want to change a lot of chord names, probably using a lot of slash chords, no. Take a simple tritone sub, but just using the bass note. An oft used ploy for bassists. The chord is C, going to an F, but towards the end of the C bar, the bassist plays an F#/Gb, to lead on to the root of F, F. That's not really a chord tone of F, and especially not of C, which is the chord in question. Is it worth calling the chord at that point in the bar anything but C?
There are plenty of other places where non-chord tones are used. Yes, the chord name could be changed, but it's only a technical exercise, which won't really help anyone.