If it were me I would go back with the receipt, don't take the coat with you and then go back, leave it there. Ask to speak with the manager or owner and be polite and say that they damaged the coat. Maybe check their website for any sort of conditions you agree to once you leave them your clothes. If they say they are not responsible then I would push back a little more forcefully and try to come to a compensation agreement.
For example, I know Terlinden wants this when a shirt has stains they could not get out with regular cleaning and one decides to have them try a better cleaning method.
Legally speaking the contract between you two is a "Werkvertrag": Terlinden are responsible for a good result and nothing else matters. In such a case Terlinden need to prove that they did everything state-of-the-art, and since they can't they need to pay, especially if you could prove that the piece was in good condition when you handed it over (e.g. because the salesperson inspected it and confirmed its status on the receipt) I doubt they could wiggle out. But if you can't make them pay you need to go to court...(see top).
Contacting the manufacturer definitely won't hurt, may actully help as he (or rather, a partner reasonably close) may be able to help determine if the piece was handled incorrectly.
There's a panel called "Paritätische Schadenerledigungsstelle"(PSE), with textiles expert(s?) among them, that you can have determin if the piece was treated properly. Costs you CHF100, no idea if their decision is binding, I'd fix that (in writing, and also fix by what amount you'd get reimbursed) before tasking them.
An alternative route(taht will likely end up with the PSE, too):
So far you appear to have nothing in your hands other than the receipt, you do need to change that. You could go there again and tape the conversation, mentioning what's happened so far (by not denying they'll implicitly confirm) and have them make all the excuses again while taping so you can show they're grasping for straw. The tape will have zero legal value (for a tape to be legally useable you must inform about the taping, and only after that may it have legal value), but it may help with your customer complaint.
When you have that produce your prepared letter that says you expect a offer from them within five (ten days if you want to give them room) business days on how they suggest to resolve this matter. Have them confirm reception on the duplicate you also brought.
After the five(ten) days plus 5 days for snail mail, contact Terlinden headquarters in ZH-Witikon.
HTH
On the other hand: I am amused by the amount of Canada Douche jackets I see in Zürich...they've become part of the Paradeplatz-Bellevue uniform, and about as appropriate in ZH as a Hummer with snow tracks...totally OTT.