Spanish:
Soy tarde
Or
Voy tarde?
Latin:
???
In Spanish:
llego tarde; I’m arriving late
vengo tarde; for some dialects and/or if you’re speaking with the person you should already have met, I am coming late
(que) (ya) voy tarde; again dialect, or when speaking to the person whose company you should already have left, “(because) I’m (already) leaving late”
voy con retraso, voy retrasado, llevo retraso “I am delayed”, mostly when you’re talking about something like a work schedule rather than about physically being somewhere
Tarde is not an adjective, it’s an adverb. Copulative verbs ser and estar go with adjectives.
I have heard some mexicans say estoy retrasado but in Spain that would mean “today I’m completely retarded”. Soy retrasado would mean “I am retarded”.
Thanks. To be more clear, I’m thinking of an illustration of the White Rabbit saying I’m late.
Have your Rabbit saying “Oh! the Duchess, the Duchess! Oh! won’t she be savage if I’ve kept her waiting! Oh my dear paws! Oh my fur and whiskers! She’ll get me executed, as sure as ferrets are ferrets!”
How do you say all that in Spanish? 
“Estoy demorado” I am delayed may be used.
“Se me hace tarde” It’s getting late (for me) should work better though.
I don’t remember very well, but in Disney’s Alice in Wonderland, translated into Spanish, I think the White Rabbit was saying “Que tarde!” (how late!) as he was running towards the hole.
«¡Oh! ¡La Duquesa, la Duquesa! ¡Cómo se pondrá si la hago esperar!» (I don’t see the rest of that in either version).
The first “I’m late” is translated «¡Dios mío! ¡Dios mío! ¡Voy a llegar tarde!»
¡Ah, la Duquesa, la Duquesa! ¡Ah! ¿No será brutal por dejarla esperando? !Ah mis pobres patas! Ah, mi piel y bigotes! ¡Me va a ejecutar, tan ciertamente como las comadrejas son comadrejas!*
I’m not a native speaker, but even I chuckle at that expression. It’s hard to describe what that would sound like in English. Something like… Lateness is an essential part of my being. But even that’s not right. It would be “I AM late” as in “I am John” or “I am human”.
A literal translation in Latin is “Sum serus”.
Oooh, missed the edit window, but I just thought of a better analogy. But you have to have watched Breaking Bad. I’m going to spoiler this because it’s such a great moment in that show:
Walt’s wife is telling him that he has to turn himself in, partly because she’s afraid the family is in danger. That there might someday be a knock on the door and… the end. Walt responds with: You have no idea who you are talking to. I am the one who knocks. I AM the danger. That first sentence was a paraphrase, but the last two are quotes.
If it meant anything, it would be something like “I am a tardy person by nature.” Or as you are phrasing it, “I am lateness” (substituting a noun for an adjective). But no one would say that.
I like that English uses “running” with “late,” which covers come and goes, Time in English often “runs,” of course, but I think “I’m running late” is used usually to say “yes, thank you very much, I know it’s late.”
Actually, which verbs languages use for time sounds interesting. I’m sure linguistics people have been on it forever…
Nm
Ontological Lateness: Merleau-Ponty’s Meta-Philosophy
Didn’t, couldn’t read it via the Webs, but I think we have a winner. 
For Latin, the Alice in Wonderland translation here (.pdf) says: “Vae, vae! Sero perveniam!” Sērō is an adverb, like Sp. tarde, and perveniam is “I will arrive,” like voy a llegar.
I’m not so sure about Gus Gusterson’s sum sērus. Edit: though later in the same translation, it has “Per aures setasque meas, quam sero sum!” Sērō is the ablative here, but I’m not sure how that works with quam—how late I am?
Well thanks all. Obviously I don’t speak either Spanish or Latin. I got the idea from this:
I thought I might do an updated version of this illustration. Maybe with Latin instead.
But you could almost say of the white rabbit that his lateness is an essential part of his being.
But in Spanish you can say soy puntual or acuse someone else of ser impuntual (for some reason the permanently tardy never seem to consider it a personal characteristic in themselves, even those who hate tardiness in others).
I don’t have it here, but the version I recall had the more literal me hará ejecutar.