I think math has more to do with your question than physics. Passage of time does not disappear at infinitesimal periods. There is a difference between infinitesimal and zero.
Infinite-small in practice means that no matter how much smaller you make it - some property will be preserved. Speed is preserved in arbitrary small valued periods of time, it also has a statistical tendency of becoming more precise (closer to the idealistic tangent one) as periods get smaller, because overall set of “time x position” choices is getting reduced on every step. Note: Although, it could depend on how those sets are defined - naive notion of precision may not always hold: error can grow for smaller periods if there were elements of a set of possible times and positions with existence conditioned on inclusion of other elements - sounds strange but actually quite common when you consider that measurement itself could be noisy/side-effectful (note: I’m still referring to classic physics here).
Example of how not just speed as a property but also its value is preserved:
Let’s say you take some period of time (let’s say a second) and start dividing it by two. No matter how many times you repeat it - you still will be able to get some non-zero number and continue.
You will not reach zero no matter how many finite steps you take. 795
You only reach zero at infinity, which I personally would say is physically impossible.
Moreover, if you take a finite interval between two positions (let’s say 1 meter) and divide it by 2 for every step (simultaneously with time) - you can notice that no matter how many steps of making them both smaller you take, the instantaneous speed would still be 1 m/s.
Note: the linear (inertial) example above is only an example of mathematical possibility: AFAIK physical methods of measuring speed do not allow such perfect circumstances (linear sync between position and time), so things can even get as complex as QM where speed has different semantics, but even classic theories rarely (or never?) talk about fully precise instant speed, as linear relationship between time and position is some artificial ideal we invented. Actual measurements mostly reject this assumption. IMO, except in computer programming where interpreting a program run as a physical experiment makes border between natural and artificial thin to the point where “physics = math” - but even digital computers are subject to noise so maybe not.
In any case, perfect linearity/inertia being an artificial construct is the practical reason why subdivisions of time periods are thought to improve precision (while natural noise actually interferes with this artificial tendency in reality) - so we could have idealized perfect inertial instant speed but only in a physically unapproachable limit, where things would look linear but really don’t :).
When it comes to other possibilities of “time erasure”, ”mathematical” time can disappear, but speed would disappear consequently too
Time does not have to be measured in seconds and position does not have to be measured in meters. Any relationship between two variables could be called speed in terms of “rate of change” like speed of pixel brightness on a still digital image along the eye tracking path.
Any order can be seen as “time” so time can disappear as order disappears (informally there is no time without “more than” comparison). But speed looses meaning without at least some notion of time, purely by definition (as “rate of change of ordered events” so on) . So the answer is still no, canonically there is no speed without time.
In other words, time can be weird and appear non-linear comparing to intuitive notion of time (think of “Pulp Fiction”) but speed cannot exist without it. 795