  We perform the first search for an isotropic non-tensorial gravitational-wave
background (GWB) allowed in general metric theories of gravity in the North
American Nanohertz Observatory for Gravitational Waves (NANOGrav) 12.5-year
data set. By modeling the GWB as a power-law spectrum, we find strong Bayesian
evidence for a spatially correlated process with scalar transverse (ST)
correlations whose Bayes factor versus the spatially uncorrelated
common-spectrum process is $99\pm 7$, but no statistically significant evidence
for the tensor transverse, vector longitudinal and scalar longitudinal
polarization modes. The median and the $90\%$ equal-tail amplitudes of ST mode
are $\mathcal{A}_{\mathrm{ST}}= 1.06^{+0.35}_{-0.28} \times 10^{-15}$, or
equivalently the energy density parameter per logarithm frequency is
$\Omega_{\mathrm{GW}}^{\mathrm{ST}} = 1.54^{+1.20}_{-0.71} \times 10^{-9}$, at
frequency of 1/year.
