Climbing a Lighthouse?

It would help a lot in my story if this were possible. But is it?

My character is near an abandoned lighthouse. It is a windy day. Never mind why (it doesn't matter for this discussion) he climbs up the side of the lighthouse--not all the way mind you, but twenty feet or so. Then he climbs back down. He is young, strong, agile. This doesn't take long for him to do.

That is the scenario. At first I kinda balked at the idea, because of the common image one has of these structures--smooth rounded towers that taper upwards. But then I googled a bit and found pics of quite different designs. Some were square, for example, and others were octagonal. Rather that a tapering rise, I saw some with levels not unlike a step pyramid. Irregular brickwork instead of smooth walls (like that of Big Sur). Others might even have a kind of middle balcony halfway up.

So in my mind, this lighthouse is at the water's edge, upon an outcrop of rock jutting out from several very narrow hills/cliffs (one of these currently is the site of a modern automated lamp serving the same function). The ten foot high base is square. Upon that rises a four-sided tower that tapers slightly upward for twenty-five feet. All of this is brick, which has eroded in irregular patterns. More, there are protruding windows (about one foot wide and five feet tall) on each side of the tower section. Atop all that is the twelve-foot-high light housing.

Does this sound like a structure a strong, agile 22-year old man could scale without too much difficulty? I'm thinking he's almost certainly done this before.

Thanks in advance.

No, he just climbs sans any gear. That is important for the plot. Just as it is important he actually climb the building, not walk up any steps.