Sleep and prolong sleep
What is it for?
Without knowing what you would use this for I can only go by your question.
The behaviour of stc is not what the  question's title suggests as it is very easy to make it fail or behave in a non intuitive manner.
Forever pending
Consider the following use example
stc.sleep(1000, &quot;test&quot;).then(()=&gt;log(&quot;Test timeout&quot;);  // this promise is never resolved
setTimeout(()=&gt;{
        stc.sleepProlong(2000, &quot;test&quot;).this(()=&gt;log(&quot;Test prolonged timeout&quot;)
    },200);

The first promise is left pending and will never be resolved.
Some other potential problems

Each time sleepProlong is called you leave another promise unresolved.
When sleepProlong is called the name implies that it &quot;prolongs sleep&quot;, but if the sleep has already timeouted out it creates a new sleep???
If sleep is called with the same name as an existing sleep, the previous sleep can no longer be prolonged,
The stc interface exposes delayTimer making the state of stc untrustable.

Expectations?
From the question's title I would expect

that there is only one promise for a named sleep.
that the sleep and prolonged sleep can be canceled
that the sleep can be prolonged indefinitely
that attempting to prolong a sleep that has timeout does nothing.
that calling sleep with the name of an unresolved existing sleep does nothing

Some general style and coding points

Function argument names are inconsistent you call the timeout delay m in one function and ms in the other.
The line delete stc.delayTimer[name]; is not needed as you overwrite the value by calling sleep on the next line.
not using curlies {} to delimited statement blocks is a bad habit, Always delimit statement blocks with {...}.You had if (stc.delayTimer[name])  clearTimeout(stc.delayTimer[name]); is better as if (stc.delayTimer[name])  { clearTimeout(stc.delayTimer[name]); }
There is no need to check if the named timeout event handle exists. clearTimeout will ignore undefined and already timed out handles. Thus if (stc.delayTimer[name])  { clearTimeout(stc.delayTimer[name]); } can be clearTimeout(stc.delayTimer[name]); and not change the behaviour.
This may just be my personal preference but it feels wrong having the time m, or ms as the first argument. Putting the name first and then the delay seems more natural (and thus easier to remember)
You don't need to await the promise you return in the async function, Async functions do this automatically .
Ensure that you protect important state variables. If entries in the  object delayTimer are mutated then you lose the ability to maintain stc's correct behaviour. You can protect it via closure.

Cleanup and keeping original behaviour
without changing the behaviour of you code I have made changes as outlined in the above points, and modified some names.
const stc = (() =&gt; {
    const handles = {};
    return {
        sleep(name, time) { return new Promise(r =&gt; handles[name] = setTimeout(r, time)) },
        async prolong(name, time) {
            clearTimeout(handles[name]);
            return stc.sleep(name, time);
        }
    };
})();