The authors transfer the common algebro-geometric method of construction of the discrete KP equation to the finite field case. First: the authors recall the discrete KP equation. Given an algebraic curve \(C\) of genus \(g\) over the finite field \(F_q\), they introduce the wave function \(\psi(n_1,n_2,n_3)\) as a rational function on \(C\) with prescribed order of zeros at certain points \(A_0,\dots,A_3\) and poles \(B_j\), \(j=1,\dots,g\). For the generic case, the \(F_q\)-valued potential \(\tau\) is defined by \(T_i\tau/ \tau=\sigma_1\), where \(T_i\) are translation operators (e.g., \(T_{1\psi} =\psi(n_1+1,n_2,n_3))\) and \(\sigma_i\) certain functions defined by the expansion of \(\psi\) at \(A_i\). Then, the discrete KP equation reads \[ (T_1 (\tau)(T_2T_3\tau)-(T_2\tau)(T_3T_1\tau)+(T_3\tau)(T_1T_2 \tau)=0. \] Second, the authors summarize the finite field version of the Krichever construction of the solution of the discrete KP equation. It is interpreted in terms of the Jacobian variety \(J(C)\). Third, the procedure is applied to the genus-two hyperelliptic curve \[ v^2+uv=u^5+5u^4+6u^2+u+3 \] over \(F_7\). Explicit results for \(n_1,n_2=1,\dots,35\) and \(n_3=-1,1,0, 2\) are stated.