The Hard Truth: If There's No Contract, They May Be Right
Under copyright law in most countries, the creator of an artwork owns it automatically — even if it was created for the band, even if the band paid for materials, even if everyone assumed it 'belonged to the band.' Without a written agreement transferring ownership, the designer/creator holds the copyright.
What a Strong Position Looks Like
If you have any of the following, you're in a stronger position: a written agreement that the work was created 'for hire' or assigned to the band, evidence that the band paid for the work as a commissioned piece, or a band agreement that states all creative work produced for the band is band property. If you have these, the leaving member's claim is much weaker.
Negotiation First
Before threatening legal action, try to negotiate. Most departing members would rather get something than get nothing and burn bridges. Offer a one-time buyout of the IP — a flat fee for full ownership transfer, documented in writing. What's it worth? That depends on how embedded the branding is. If you have 10,000 Instagram followers and a recognisable logo, it's worth paying a fair fee to own it cleanly.
If They Won't Budge
You have two practical options. Option 1: Commission a redesign. It hurts in the short term but gives you clean IP going forward. Treat it as a rebranding exercise. Option 2: Get a lawyer to assess whether 'work made for hire' or implied licence arguments apply in your jurisdiction. In some cases, even without a written agreement, a court will recognise that the work was created in the context of a working relationship that implies certain usage rights.
Prevent It Next Time
Add an IP assignment clause to your band agreement: all creative work produced for the band — logos, artwork, website design, photography — is owned by the band as an entity, not by the individual who created it. Have this signed before anyone creates anything.