  The recent detection of a stochastic signal in the NANOGrav 15-year data set
has aroused great interest in uncovering its origin. However, the evidence for
the Hellings-Downs correlations, a key signature of the gravitational-wave
background (GWB) predicted by general relativity, remains inconclusive. In this
letter, we search for an isotropic non-tensorial GWB, allowed by general metric
theories of gravity, in the NANOGrav 15-year data set. Our analysis reveals a
Bayes factor of approximately 2.5, comparing the quadrupolar (tensor
transverse, TT) correlations to the scalar transverse (ST) correlations,
suggesting that the ST correlations provide a comparable explanation for the
observed stochastic signal in the NANOGrav data. We obtain the median and the
$90\%$ equal-tail amplitudes as $\mathcal{A}_\mathrm{ST} = 7.8^{+5.1}_{-3.5}
\times 10^{-15}$ at the frequency of 1/year. Furthermore, we find that the
vector longitudinal (VL) and scalar longitudinal (SL) correlations are weakly
and strongly disfavoured by data, respectively, yielding upper limits on the
amplitudes: $\mathcal{A}_\mathrm{VL}^{95\%} \lesssim 1.7 \times 10^{-15}$ and
$\mathcal{A}_\mathrm{SL}^{95\%} \lesssim 7.4 \times 10^{-17}$. Lastly, we fit
the NANOGrav data with the general transverse (GT) correlations parameterized
by a free parameter $\alpha$. Our analysis yields
$\alpha=1.74^{+1.18}_{-1.41}$, thus excluding both the TT ($\alpha=3$) and ST
($\alpha=0$) models at the $90\%$ confidence level.
