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The recently proposed novel materials class called Weyl-Kondo semimetal (WKSM) is a time reversal invariant but inversion symmetry broken Kondo semimetal in which Weyl nodes are pushed to the Fermi level by the Kondo interaction. Here we explore whether CeCu$_{1+x}$As$_2$ may be a new WKSM candidate. We report on its single-crystal growth, structure determination and physical properties investigation. Previously published studies on polycrystalline samples suggest that it is indeed a Kondo semimetal, which is confirmed by our investigations on single crystals. X-ray diffraction reveals that CeCu$_{1+x}$As$_2$ crystallizes in a tetragonal centrosymmetric structure, although the inversion symmetry could still be broken locally due to partially occupied Cu sites. Chemical analysis results in an average occupation $x$ = 0.11(1). The electrical resistivity increases logarithmically with decreasing temperature, and saturates below 10 K. A Kondo temperature $T_{\mathrm{K}}$ $\approx$ 4 K is extracted from entropy, estimated from the specific heat measurements. From Hall effect experiments, a charge carrier density of $8.8 \times 10^{20}$ cm$^{-3}$ is extracted, a value characteristic of a semimetal. The magnetization shows pronounced anisotropy, with no evidence of magnetic ordering down to 0.4 K. We thus classify CeCu$_{1.11}$As$_2$ as a tetragonal Kondo semimetal with anisotropic magnetic properties, with a possibly broken inversion symmetry, thus fulfilling the necessary conditions for a WKSM state.