Nonlinear evolution equations (NLEEs) are seen in such fields as fluid dynamics, plasma physics and optics. A (3+1)-dimensional time-dependent-coefficient Boiti-Leon-Manna-Pempinelli equation is investigated in this paper. Via the logarithmic transformation on non-zero background, a bilinear form is derived. Via the bilinear form, a bilinear-Bäcklund transformation with some solutions is acquired, while the one-soliton, two-soliton and multiple soliton solutions with two different nonlinear dispersion relations are worked out. On some non-zero backgrounds, multi-kink solutions are derived. Via the complex conjugation, half-periodic kink solutions are obtained. •A (3+1)-dimensional time-dependent-coefficient Boiti-Leon-Manna-Pempinelli equation is investigated in this paper.•Via the Hirota method, a bilinear form of that equation is derived.•A set of bilinear Bäcklund transformations have been obtained.•Soliton solutions on non-zero backgrounds with two cases of nonlinear dispersion relations are derived.•With some backgrounds, multi-kink and half-periodic kink solutions are derived.