I assume "spectre-like" means "leaks info due to side effects of speculative execution". Arm (and any high performance chip really) relies on speculative execution for performance, so I would be surprised if they are not susceptible to spectre-like attacks in principle. What am I missing?

Nothing.

At least for now, the fastest ARM CPUs are not immune to Spectre, so the recent ARM cores have introduced a set of similar workarounds to the recent Intel and AMD CPUs, in order to serialize the execution and flush internal state around context changes.

Searching for "Speculative Processor Vulnerability" on the Arm developer site will find many resources describing possible attacks and the mitigations that must be implemented for various Arm cores.

But this isn't one. Every modern design that's used in any capacity turned out to be vulnerable one way or another.