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] <cite authors="Zettl, Heinrich">_Zettl, Heinrich_, [**A characterization of ternary rings of
operators**](http://dx.doi.org/10.1016/0001-8708(83)90083-X), Adv. Math. 48, 117-143
(1983). [ZBL0517.46049](https://zbmath.org/?q=an:0517.46049).</cite>