"No" it is has only unit (steradian) not DF

Approach=

Step 1. formula-   solid angle is the ratio of area of segment to the square of radius of the sphere of which  the segment  is a part

Step 2. DF of each paramter in the formula, as area and square of radius both are a type of length square or in other words "both" have dimension of length square so their ratio becomes dimensionless.