Given parameters $α\geq 1,β\geq 0$, a subgraph $G'=(V,H)$ of an $n$-vertex unweighted undirected graph $G=(V,E)$ is called an $(α,β)$-spanner if for every pair $u,v\in V$ of vertices, $d_{G'}(u,v)\leq αd_{G}(u,v)+β$. If $β=0$ the spanner is called a multiplicative $α$-spanner, and if $α= 1+ε$, for an arbitrarily small $ε>0$, the spanner is said to be a near-additive one. Graph spanners are a fundamental and extremely well-studied combinatorial construct, with a multitude of applications in distributed computing and in other areas. Near-additive spanners, introduced in [EP01], preserve large distances much more faithfully than multiplicative spanners. Also, recent lower bounds [AB15] ruled out the existence of arbitrarily sparse purely additive spanners (i.e., spanners with $α=1$), and therefore near-additive spanners provide the best approximation of distances that one can hope for. Numerous distributed algorithms for constructing sparse near-additive spanners exist. In particular, there are now known efficient randomized algorithms in the CONGEST model that construct such spanners [EN17], and also there are efficient deterministic algorithms in the LOCAL model [DGPV09]. The only known deterministic CONGEST-model algorithm for the problem [Elk01] requires superlinear time in $n$. We remedy the situation and devise an efficient deterministic CONGEST-model algorithm for constructing arbitrarily sparse near-additive spanners. The running time of our algorithm is low polynomial, i.e., roughly $O(β\cdot n^ρ)$, where $ρ> 0$ is an arbitrarily small positive constant that affects the additive term $β$. In general, the parameters of our algorithm and of the resulting spanner are at the same ballpark as the respective parameters of the state-of-the-art randomized algorithm for the problem due to [EN17].