Yes, from the server's invite page — there is a Pause control under your subscription. Billing stops and your Discord role comes off until you resume, which is the part worth knowing before you press it.
On the server’s flatpass invite page, under your subscription, while you are signed in. It is not on Stripe’s billing page, which only offers cancelling; the Pause control is on the invite page.
Pausing asks you to confirm, because the role comes off the moment you press it and “pause billing” does not sound like that. Resuming is a single press on the same page and needs no confirmation.
Billing stops. No charge is taken while you are paused, and the months you miss are not banked to be collected later — there is no bill waiting for you when you come back.
Your Discord role comes off. That is the trade: you are not paying, so you do not have the access. The server’s invite page says plainly that you are paused, which is otherwise something you would only notice by the role being gone.
You keep your plan and your price. It is the same subscription, held rather than ended. Resume and you go back to what you were on — even if the seller has raised the price since. Cancelling loses that.
You can still open the billing page while paused, to read invoices or update the card on file.
During a free trial. Trial days run down against a fixed date and pausing does not move it, so a paused trial would take your role away while the free days drained anyway. You would come back to an expired trial you never used, and a charge. The invite page says when the control returns rather than hiding it.
Run /membership in the server and press Manage billing, or use the manage link on the server's invite page. The role stays until the period you've paid for ends.
Run /membership, press Manage billing, and update the payment method in Stripe's portal. A failed renewal keeps your role while Stripe retries.
Run /membership in the server: it re-applies the role on the spot. If it can't, the server's owner has one setting to change and has already been told.