I regard a standard panel study as a two -level multilevel structure with repeated measures at level 1 nested within individuals at level 2. Gender is level 2 variable that is time invariant and will as such always be excluded in a fixed effects analysis. Or to put it another way the fixed dummies have soaked up all the cross sectional variation and the is nothing left to estimate a parameter associated with a time invariant variable. This is exact collinearity, you cannot escape it in a fixed effects analysis.

However, you can get best of both worlds by using random effects in its within between form. You can estimate a cross sectional effect ( in your case Gender) and get same result for time varying variables as if you used fixed effects, if you include their cross sectional mean.See my recent publications with Andy Bell and Malcolm Fairbrother for the arguments.

Article Explaining Fixed Effects: Random Effects Modeling of Time-Se...

Article Fixed and Random effects models: making an informed choice

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The Hausman test is a red herring, it is telling you that there is a cross sectional effect for a time varying variable.

We are aware that these views are somewhat controversial!

see this Wikipedia entry https://en.wikipedia.org/wiki/Kelvyn_Jones

Here is an extract

"He (with colleagues) has challenged the 'gold standard' that fixed effects should be the standard approach to the analysis of Panel data and that a Hausman test is an appropriate way of choosing between a Fixed effects model and a Random effects model. Somewhat controversially they argue that a particular form of the random effects model (the within-between model or the similar Mundlak model) offers all that fixed effects can provide and more.[16][17][18] They also challenge the Fixed Effects Vector Decomposition (FEVD) model of Plumper and Troeger.[19] One reaction was: "This paper and the instructive controversial over FEVD have shown me that my econometrics training had not - as I once assumed - taught me all that there is to know about fixed effects estimation. In particular, the authors' treatment of 'heterogeneity bias' clarifies the importance of addressing both 'within' and 'between' variation in the data and they make a compelling case for considering both 'individual' and 'ecological' influences".[20] Another was: "Bizarre and often incorrect paper by two political scientists on the virtues of random-effects over fixed-effects".[21] to "You can and should use a well-specified random effects model. Always.".[22] "

and another

"He and colleagues argue that group-mean centering in multilevel models can be a useful procedure in random coefficient models,[23] thereby disagreeing that it is a 'dangerous' procedure.[24] Reactions to this critique include "may the Saints & Angels protect us from ever having a paper this thoroughly dismantled"[25] and "Seriously though, if you are interested in multilevel modelling I highly recommend this short, instructive and frankly rather sassy paper." [26] "

i have same problem too

Muhammad Arafiq : As Kelvyn Jones mentioned above, gender is a time-invariant variable (in most cases). Therefore, time-invariant variables will always be excluded in a FE analysis.