Arduino nano digital I/O overload
Short answer is probably.  There is a voltage drop from base to emitter that is about .7 volts.  But something still has to contend with the rest of the energy.  You might try a different Arduino digital output on the chance that the balance of the processor is working as expected.
I say probably because several things might have happened if the transistor actually turned on.  For instance the forward biased (backwards) diode may have allowed so much current to flow through the relay and transistor that the transistor failed first.  Or, if the power supply had a low current rating, it may not have had the capacity to burn anything out.  As you can see, it is difficult to say what exactly happened with out testing each part individually.