BPSK signal that has not been pulse shaped by any filter
To answer your question, NO!  A BPSK signal with a random data bit pattern that has not gone through any (additional) pulse shaping, meaning the pulse shape is rectangular in time equal to the bit duration, has a $Sinc^2$ power spectral density centered on the carrier frequency (as the Fourier Transform of a rectangular pulse is the Sinc function).

This signal has infinite bandwidth, but the power spectral density is dropping within an envelope of $1/f^2$ where f is the offset from the carrier (as expected from the $Sinc^2$ function). Any reduction of this bandwidth by filtering in the frequency domain, results in a finite transition time between symbols, so therefore induces a pulse shape in the time domain that is non-rectangular. 

Another way to see this simply, a symbol that can transition from one state to the next in an infinitely short time (which is what the rectangular pulse shape is required to do) would require infinite bandwidth in order to transition that quickly. We slow down the transition rate on purpose in order to limit the bandwidth occupied since spectrum is such a valuable (and regulated) resource. We just do that very strategically to minimize inter-symbol interference, and optimize the SNR we can receive within the bandwidth used.