  Cosmic-ray interactions with the solar atmosphere are expected to produce
particle showers which in turn produce neutrinos from weak decays of mesons.
These solar atmospheric neutrinos (SA$\nu$s) have never been observed
experimentally. A detection would be an important step in understanding
cosmic-ray propagation in the inner solar system and the dynamics of solar
magnetic fields. SA$\nu$s also represent an irreducible background to solar
dark matter searches and a detection would allow precise characterization of
this background. Here, we present the first experimental search based on seven
years of data collected from May 2010 to May 2017 in the austral winter with
the IceCube Neutrino Observatory. An unbinned likelihood analysis is performed
for events reconstructed within 5 degrees of the center of the Sun. No evidence
for a SA$\nu$ flux is observed. After inclusion of systematic uncertainties, we
set a 90\% upper limit of
$1.02^{+0.20}_{-0.18}\cdot10^{-13}$~$\mathrm{GeV^{-1}cm^{-2}s^{-1}}$ at 1 TeV.
