create a sequential number for each group

A simple solution with Base R:

df$seq <- ave(sapply(df$gap, identical, "gap"), df$id, FUN = cumsum)
df
#>          id                date      lc     lon     lat   gap_days  gap seq
#> 1  20162.03 2003-10-19 14:33:00 Tagging -39.370 -18.480         NA <NA>   0
#> 2  20162.03 2003-10-21 12:19:00       1 -38.517 -18.253 1.90694444  gap   1
#> 3  20162.03 2003-10-21 13:33:00       1 -38.464 -18.302 0.05138889   no   1
#> 4  20162.03 2003-10-21 16:38:00       A -38.461 -18.425 0.12847222   no   1
#> 5  20162.03 2003-10-21 18:50:00       A -38.322 -18.512 0.09166667   no   1
#> 6  20162.03 2003-10-23 10:33:00       B -38.674 -19.824 1.65486111  gap   2
#> 7  20162.03 2003-10-23 17:52:00       B -38.957 -19.511 0.30486111   no   2
#> 8  20162.03 2003-11-02 08:14:00       B -42.084 -24.071 9.59861111  gap   3
#> 9  20162.03 2003-11-02 09:36:00       A -41.999 -24.114 0.05694444   no   3
#> 10 20687.03 2003-10-27 17:02:00 Tagging -39.320 -18.460         NA <NA>   0
#> 11 20687.03 2003-10-27 19:44:00       2 -39.306 -18.454 0.11250000   no   0
#> 12 20687.03 2003-10-27 21:05:00       1 -39.301 -18.458 0.05625000   no   0

And then split it:

split(df, list(df$id, df$seq), drop = TRUE)
#> $`20162.03.0`
#>         id                date      lc    lon    lat gap_days  gap seq
#> 1 20162.03 2003-10-19 14:33:00 Tagging -39.37 -18.48       NA <NA>   0
#> 
#> $`20687.03.0`
#>          id                date      lc     lon     lat gap_days  gap seq
#> 10 20687.03 2003-10-27 17:02:00 Tagging -39.320 -18.460       NA <NA>   0
#> 11 20687.03 2003-10-27 19:44:00       2 -39.306 -18.454  0.11250   no   0
#> 12 20687.03 2003-10-27 21:05:00       1 -39.301 -18.458  0.05625   no   0
#> 
#> $`20162.03.1`
#>         id                date lc     lon     lat   gap_days gap seq
#> 2 20162.03 2003-10-21 12:19:00  1 -38.517 -18.253 1.90694444 gap   1
#> 3 20162.03 2003-10-21 13:33:00  1 -38.464 -18.302 0.05138889  no   1
#> 4 20162.03 2003-10-21 16:38:00  A -38.461 -18.425 0.12847222  no   1
#> 5 20162.03 2003-10-21 18:50:00  A -38.322 -18.512 0.09166667  no   1
#> 
#> $`20162.03.2`
#>         id                date lc     lon     lat  gap_days gap seq
#> 6 20162.03 2003-10-23 10:33:00  B -38.674 -19.824 1.6548611 gap   2
#> 7 20162.03 2003-10-23 17:52:00  B -38.957 -19.511 0.3048611  no   2
#> 
#> $`20162.03.3`
#>         id                date lc     lon     lat   gap_days gap seq
#> 8 20162.03 2003-11-02 08:14:00  B -42.084 -24.071 9.59861111 gap   3
#> 9 20162.03 2003-11-02 09:36:00  A -41.999 -24.114 0.05694444  no   3