Physical properties of the quasi-two-dimensional square lattice antiferromagnet <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Ba</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi>FeSi</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="normal">O</mml:mi><mml:mn>7</mml:mn></mml:msub></mml:mrow></mml:math>

We report magnetization (\ensuremath{\chi}, $M$), magnetic specific heat (${C}_{\mathrm{M}}$), and neutron powder diffraction results on a quasi-two-dimensional (2D) $S=2$ square lattice antiferromagnet ${\mathrm{Ba}}_{2}{\mathrm{FeSi}}_{2}{\mathrm{O}}_{7}$ consisting of ${\mathrm{FeO}}_{4}$ tetrahedrons with highly compressive tetragonal distortion (27%). Despite the quasi-2D structure, both \ensuremath{\chi} ${C}_{\mathrm{M}}$ present three-dimensional long-range ordering below N\'eel temperature ${T}_{\mathrm{N}}=5.2\phantom{\rule{0.16em}{0ex}}\mathrm{K}$. Neutron data show collinear ${\mathbit{Q}}_{\mathbit{m}}=$(1,0,1/2) antiferromagnetic (AFM) structure ${T}_{\mathrm{N}}$ but ordered moment aligned in $ab$ plane is suppressed by 26% from ionic spin value ($4{\ensuremath{\mu}}_{\mathrm{B}}$). Both AFM moments are well explained using Monte Carlo simulations large single-ion in-plane anisotropy $D=1$.4 meV rather small Heisenberg exchange ${J}_{\mathrm{intra}}=0$.15 plane. The characteristic 2D fluctuations recognized entropy release diffuse scattering above ${T}_{\mathrm{N}}$. This new system also displays unusual nonmonotonic dependence as function field $H$.