Considering you have a pretty large dataset, I think using data.table will be better ! Here is the data.table version to solve your problem, it will be quicker than dplyr package:
library(data.table)
df <- data.table(ID=c("A","A","A","B","B","C"),c1=1:6,c2=1:6,
myDate=c("01.01.2015","02.02.2014",
"03.01.2014","09.09.2009","10.10.2010","06.06.2011"))
df[,myDate:=as.Date(myDate, '%d.%m.%Y')]
> df_new <- df[ df[, .I[myDate == min(myDate)], by=ID]$V1 ]
> df_new
ID c1 c2 myDate
1: A 3 3 2014-01-03
2: B 4 4 2009-09-09
3: C 6 6 2011-06-06
PS: you can use setDT(mydf) to transform data.frame to data.table.
