If you are constrained to move in a plane, then yes, circular motion is the only possibility. To see this, let (x(t),y(t)) be the coordinates at time t on the path. The condition is

$x\frac{dx}{dt}+y\frac{dy}{dt}=0$

Integrate with respect to t:

$\int x\frac{dx}{dt}dt+\int y\frac{dy}{dt}dt=\int 0,dt$

$\frac{{x}^{2}}{2}+\frac{{y}^{2}}{2}=C$

which is the equation of a circle.

So your path has to move around the circle. But that doesn't mean the path itself is a circle. It can speed up. It can slow down. It can stop and reverse direction, all without violating the perpendicularity condition (when it is stopped, the velocity is 0, whose inner product with the position vector is still 0 - this stretches the definition of "perpendicular" a bit, but generally it's allowed, as the physical objects being modelled occasionally do stop).

In three dimensions, a similar calculation shows you that the motion is constrained to some sphere. But you can move in all sorts of ways on a sphere that do not qualify as "circular".