Just a few years ago, Europe’s banks managed to stagger out of a crisis brought on by the continent’s debt woes. Britain’s looming exit from the EU, analysts and investors fear, could push them back in.

A wide swathe of European financial institutions are at risk: hobbled behemoths like Deutsche Bank and Credit Suisse that are limping through difficult turnarounds, clusters of regional banks pressured by negative interest rates, and banks across Europe's weak periphery that are reeling under piles of bad loans.