I have a RasterBrick consisting of monthly rainfall data over 7 years, so it has 7 layers with 12 slots each:
rainfall <- brick(\"Rainfall.tif\")
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Each layer in your RasterBrick will have a unique name, so you can use match('name', names(b)) to find the numeric index of the layer you're interested in. Then use the layer= argument to extract() to point to the layer from which you'd like to extract (setting nl=1 to indicate that you only want that one layer).
Here's a reproducible example, in which I use cell numbers to do the extracting. (This will work exactly the same way when using SpatialPoints to indicate which values are to be grabbed.)
## An example SpatialBrick
b <- brick(system.file("external/rlogo.grd", package="raster"))
nlayers(b)
# [1] 3
names(b)
# [1] "red" "green" "blue"
## Extract data from given cells in the "green" layer,
ii <- match("green", names(b))
extract(b, 1000:1003, layer=ii, nl=1)
# green
# [1,] 254
# [2,] 255
# [3,] 255
# [4,] 255