Dictionaries would indicate that the correct form is pick up, and not pick, and that the shortened usage is not the right way to say it. Put another way, if it's becoming acceptable (and language does evolve, so there's always a chance that it will), than the dictionaries I consulted haven't acknowledged so, as of yet.
For example, I looked at Macmillan's1 entry for pick, and found that – at least according to that source – you can pick your friends, and you can pick your guitar, but you can't pick your friends at the airport.2
More explicitly, Macmillan lists four definitions for pick: (1) to choose or select; (2) to take with your fingers; (3) to pull at with your fingertips or fingernails, i.e., pick a scab; (4) to pluck a guitar. There are also several phrases listed with the word pick: pick and choose, pick someone's brain, pick a fight, pick holes in an argument, pick a lock, pick a nose, pick a pocket, pick your teeth, pick something apart, pick your way through someplace, and pick a winner.
Morever, the OED adds some other meanings for the verb pick: to use or wield a pick or pickaxe, to pluck a fowl, to eat bit by bit, to open a lock, to take or steal, to acquire or gain, to pick at with a beak (said of a bird), to find an opportunity for a quarrel (pick a fight), to strike up a conversation, to guess or deduce, to read a bowler (in cricket), to shoe a horse (rare).
In short, I can't find a definition of the word pick that would allow it to be used like this: I will pick my friend at the airport tomorrow, not if the speaker was referring to going to the airport to meet a friend, and then drive that friend to a hotel or something.
However, Macmillan's dictionary does have a phrasal entry for pick up, and meaning #2 reads:
to go and meet someone or something that you have arranged to take
somewhere in a vehicle
Similarly, the OED, under entry 3b for the phrasal verb pick up reads:
To take (a person or thing) into one's company; (in later use) esp. to stop for (a person) to get into a vehicle.
So, to answer your question, I would dub the shortened form as either (c) erroneous, or perhaps (b) local slang. For now, the dictionaries would back me up.
All that said, I wonder... Macmillan's pick up entry also has meaning #10, which reads:
to start talking to someone because you want to have sex with them
Similarly, in the OED, entry 3c:
Originally: to engage the sexual services of (a prostitute, etc.), to proposition. Now chiefly: to strike up a relationship with (a person, esp. a stranger) as a sexual overture; (more generally) to form an informal or casual acquaintance with.
That got me thinking: what if some non-natives, when they were learning English, were told something along the lines of, "Be careful how you use the words 'pick up'! When you say you're trying to pick someone up, it can mean that you want to sleep with them." That might explain why you run across the abbreviated form from time to time; I can understand being reluctant to use the more correct wording, especially if one was wary of creating an unintentional double entendre.
That is only a theory.
1I've linked to the British edition, but the American edition had the same results.
2Just to be clear, this was a play on the oft-quoted pun found here.