According to Zettl [1], a ternary ring of operators (TRO) is a ternary $C^*$-ring which is isomorphic to a closed subset $X\subseteq B(H)$, such that $XX^*X\subseteq X$, and is equipped with the ternary multiplication $$ [x,y,z] := xy^*z. $$ On the other hand, an anti-TRO is a ternary $C^*$-ring defined as above, except that the multiplication operation is $$ [x,y,z] := -xy^*z. $$ It is a fundamental result of Zettl [1] that every ternary $C^*$-ring $X$ decomposes uniquely as $$ X=X_+\oplus X_-, $$ where $X_+$ is a TRO, and $X_-$ is an anti-TRO .
It seems to me that the reading of the question posed by the OP that makes the most sense is by taking the expression "operator space" to mean a TRO. In this case the answer is yes, there does exist a ternary $C^*$-ring which is not an TRO: just take any non-zero anti-TRO. For an even more concrete example, take $X=M_{n\times m}({\bf C})$, with ternary multiplication $[x,y,z] := -xy^*z$.
On the other hand, if one takes the expression "operator space" for its face value, Zettl's result implies that every ternary $C^*$-ring is an operator space in an even more canonical form than suggested by user @YemonChoi: write $X=X_+\oplus X_-$, embedd $X_+$ in $B(H_+)$, and $X_-$ in $B(H_-)$, whence $$ X\subseteq B(H_-\oplus H_+). $$ This embeding preserves the operator space structure (norms on matrix algebras) that a TRO canonical possesses.
It is interesting to remark that if you change the (binary) multiplication operation on a $C^*$-algebra by $$ x\circ y := -xy, $$ then the resulting object is strictly speaking a new C*-algebra, but it is isomorphic to the old one. The isomorphism is simply $a\mapsto -a$.
However, if you change the (ternary) multiplication on a ternary $C^*$-ring by inserting a minus sign as above, then the map $a\mapsto -a$ is no longer an isomorphism, essentially because 2 is even and 3 is odd! Indeed, Zettl's uniqueness result tells you that the new ternary $C^*$-ring might not be isomorphic to the old one at all!
[1] Zettl, Heinrich, A characterization of ternary rings of operators, Adv. Math. 48, 117-143 (1983). ZBL0517.46049.