Is the speed of light the only speed that exists?

The norm of the four velocity is always $c$, but of course not all four velocities are equal because they can point in different directions and vectors that point in different directions are not identical.

But to claim the word speed means the norm of the four-velocity seems unjustified. By speed we normally mean (the magnitude of) the coordinate velocity, and the coordinate velocity is the derivative of position in our coordinate system with time in our coordinate system. This can of course have any magnitude from zero up to the speed of light.

For the velocity four-vector $\eta^{\mu}=\frac{dx^{\mu}}{d\tau}$ [where $d\tau=\frac{dt}{\gamma}$ stands for the infinitesimal proper time, and $x^{\mu}=(ct,x,y,z)$; you can define $\eta^{\mu}$ also as $\frac{dx^{\mu}}{dt}$; both have their advantages and I'm not sure which one is the commonest used but I use the first definition here] we have:

$\eta^{\mu}=\gamma(c,\vec v)$, with $\gamma=\frac 1 {\sqrt{{1-\frac {v^2} {c^2}}}}$.

So even if we stand still ($\vec v=0$, with respect to the Universe) we move with velocity (speed) $c$, which is not a vector (its the time component of the velocity four-vector, but nevertheless, its "direction" is the future).

Now $\eta_{\mu} \eta^{\mu}=c^2$ (you can calculate this yourself if you're familiar with contra- and covariant four-vectors), so the value of the velocity four-vector is always $c$.

If we stand still (but "moving" with speed $c$ through time) and start to move, a lesser portion of ${\eta}^{\mu}$ is moving through time and a $\vec v$ through space, which can have every direction, develops. The faster we move the lesser is the portion of $\eta^{\mu}$ moving through time and the bigger the portion moving through space. If we (almost) reach the velocity (through space) $\vec c$, we are (almost) only moving through space and (almost) no longer through time (as observed by someone who stays behind where we started moving). The photon is a good example of this. It moves through space with velocity $\vec c$, for all observers, while its velocity (speed) through time is zero for all observers (for a photon there is no time ticking away).

Thus every object is or moving with speed $c$ through time (directed to the future) when standing still, as strange as this may seem, or moving with speed ${\gamma}c$ through time and velocity $\gamma{\vec v}$ through space (in whatever direction) with $\eta_{\mu}\eta^{\mu}=c^2$, or with speed $0$ through time and velocity $\vec c$ through space (in whatever direction).

Maybe you can see a difficulty arising when $v=c$ because in that case $\gamma\rightarrow\infty$, giving a problem for $\eta^{\mu}=\gamma(c,\vec{v})$, but $\eta_{\mu}\eta^{\mu}$ always equals $c^2$ (actually it's a bit more complicated than that; in the linked article you can read this is connected to an affine parameter, but let's not go into thát before it's gétting too complicated!).

My answer incorporates features of the earlier answers, but tries to clearly disentangle the implied uses of "speed" in space and in spacetime [when a phrase like 'slower in the dimension of time' is used].

"Massive objects (like a basketball) all move at constant 'speed' c in spacetime" really means that

• its 4-momentum vector can be normalized to a "unit" 4-velocity vector with Minkowski-norm c.
• It has a spatial speed (slope) v < c because the spatial component of its 4-momentum has an absolute-size that is smaller than that of the time-component of its 4-momentum.

"Massless objects (like a light-signal) all move at constant 'speed' zero in spacetime" really means that

• its 4-momentum vector has Minkowski-norm zero [and thus can't be normalized].

• It has a spatial speed (slope) c because the spatial component of its 4-momentum has the same absolute-size as that of the time-component of its 4-momentum.