Engine stalling, dashboard instruments not working

My car stalls when idling. If I try to restart it, the key makes a clicking noise, but the engine does not react. If I come back later, it can start again.

The dashboard instruments do not work, but before they stopped working the battery warning light was briefly on.

I assume that the battery is busted, but I was wondering if it could be the alternator instead. Is there any way to tell based on the symptoms?