The following will return the lowest number whether 0 or negative
and the poster definitely indicated no zeros.
=SMALL(A:A,COUNTIF(A:A,0)+1)
If there are no numbers in the column
the MIN formula returns #NUM!
the SMALL formula returns 0 -- zero
Reminder the MIN formula -- entered as array formula Ctrl+Shift+Enter
=MIN(IF(A1:A200>0,A1:A200))
IF you don't want to see the error you could code use Ctrl+Shift+Enter
=IF(COUNTIF(A1:A200,">0"),MIN(IF(A1:A200>0,A1:A200)),"")