How do I deal with a belligerent colleague?

I thought I'd ask this on FDS since a lot of you have excellent advice for levelling up in life outside of dating as well so here goes.

I am in a program and amidst my cohort there are only 2 other girls. The majority are men. However, my main source of distress lately has been due to one of these women (shockingly). Before I detail these issues I want the advice to be tailored towards me looking like the levelheaded/smart or overall more mature one.

Anyways so this person (I'll just call her Jessica from here on out), though she is obviously a bit challenged

• considers herself nonbinary but only about 1/4 of the time and if you don't guess correctly she gets mad

• has openly bragged about how many psych ward trips she has gone on, how many times she has attempted to end her own life, and how many mental illnesses she has [ADHD, autism, bipolar 1 & 2, I can't remember the rest]

• openly fetishizes gay anime men who look more like underaged boys despite being a lesbian

• is edging close to 30 years old and has that yaoi obsession like I mentioned previously

All that to give you a sort of more cohesive picture of who I'm dealing with. With the election results having come about, obviously the other girl and I (we are both white [and furthermore both of us voted for Kamala]) were feeling a bit disheartened. Jessica proclaims to us that she did not vote, and that half of her family is getting deported (she is half white, half Hispanic), but also that we as white women basically need to suck it up because our rights are not going to be affected in the slightest (we are all in a very conservative red state in the southern US). Jessica is also in a bunch of online leftist groups, of what sway I'm not entirely sure.

This is just the latest in a series of incidents I have too many examples of to mention, but I'll just give a few poignant ones.

• I buy new shoes, wear them to campus. She asks me if they are used (they're not) and tells me I got ripped off, proceeds to manhandle my foot to 'inspect' the shoes. I was just so dumbfounded I didn't have anything really to say.

• Randomly got into an argument with me about if a tv show was made by a pedophile (???). It wasn't and she refused to listen to me until I had to Google it to show her.

• Drew a caricature of me as that Sigma male super muscular dude (the ones from the memes) and left it pinned up in my desk space as a joke. (I know this sounds ridiculous but it's true)

• Gets angry at me when I spend money on things. The third girl in our cohort invited us out to go jean shopping and I bought some jeans. Jessica said I could buy anything I wanted since I had 'Daddy's money' when I used my own money to complete this purchase.

In sum, how should I deal with this? Thus far I have just been being silent and keeping my expressions neutral (which I think frustrates her since she can't figure out what I'm thinking) but I've received some advice to basically just return her energy and be rude back, but I feel that's not very classy and may make me look bad in front of other colleagues. Any advice from women in academic fields or otherwise would be greatly appreciated.